A Comprehensive Review on the Generalized Sylvester Equation AX − YB = C

Abstract

Since Roth’s work on the generalized Sylvester equation (GSE) $AX-YB=C$ in 1952, related research has consistently attracted significant attention. Building on this, this review systematically summarizes relevant research on GSE from five perspectives: research methods, constrained solutions, various generalizations, iterative algorithms, and applications. Furthermore, we provide comments on current research, put forward several intriguing questions, and offer prospects for future research trends. We hope this work can fill the gap in the review literature on GSE and offer some inspiration for subsequent studies in the field.

Contents
1.	Introduction.................................................................................................................................................................................................	3
2.	Preliminaries...............................................................................................................................................................................................	4
3.	Roth’s Equivalence Theorem......................................................................................................................................................................................................	6
4.	Different Methods on GSE...............................................................................................................................................................................................................	7
	4.1.      Method by Linear Transformations and Subspace Dimensions................................................................................................	7
	4.2.      Method by Generalized Inverses...................................................................................................................................................	8
	4.3.      Method by Singular Value Decompositions.................................................................................................................................	10
	4.4.      Method by Simultaneous Decompositions...................................................................................................................................	12
	4.5.      Method by Real (Complex) Representations................................................................................................................................	13
	4.6.      Method by Determinable Representations...................................................................................................................................	15
	4.7.      Method by Semi-Tensor Products..................................................................................................................................................	20
5.	Constrained Solutions of GSE..................................................................................................................................................................	24
	5.1.      Chebyshev Solutions and lp-Solutions..........................................................................................................................................	25
	5.2.      ★-Congruent Solutions...................................................................................................................................................................	26
	5.3.      (Minimum-Norm Least-Squares) Symmetric Solutions...............................................................................................................	27
	5.4.      Self-Adjoint and Positive (Semi)Definite Solutions......................................................................................................................	28
	5.5.      Per(Skew)Symmetric and Bi(Skew)Symmetric Solutions.............................................................................................................	29
	5.6.      Maximal and Minimal Ranks of the General Solution..................................................................................................................	31
	5.7.      Re-(Non)negative and Re-(Non)positive Definite Solutions.........................................................................................................	32
	5.8.      η-Hermitian and η-Skew-Hermitian Solutions..............................................................................................................................	35
	5.9.      ϕ-Hermitian Solutions......................................................................................................................................................................	37
	5.10.      Equality-Constrained Solutions.......................................................................................................................................................	38
6.	Various Generalizations of GSE.................................................................................................................................................................	40
	6.1.      Generalizing RET over Different Rings...........................................................................................................................................	40
	6.1.1.      Generalizing RET over Unit Regular Rings......................................................................................................................	40
	6.1.2.      Generalizing RET over Principal Ideal Domains..............................................................................................................	41
	6.1.3.      Generalizing RET over Division and Module-Finite Rings...........................................................................................	42
	6.1.4.      Generalizing RET over Commutative Rings...................................................................................................................	43
	6.1.5.      Generalizing RET over Artinian and Noncommutative Rings.....................................................................................	44
	6.2.      Generalizing RET to a Rank Minimization Problem....................................................................................................................	45
	6.3.      GSE over Dual Numbers and Dual Quaternions..........................................................................................................................	46
	6.4.      Linear Operator Equations on Hilbert Spaces...............................................................................................................................	49
	6.5.      Tensor Equations...............................................................................................................................................................................	52
	6.6.      Polynomial Matrix Equations..........................................................................................................................................................	56
	6.6.1.      By the Divisibility of Polynomials.....................................................................................................................................	56
	6.6.2.      By Skew-Prime Polynomial Matrices................................................................................................................................	57
	6.6.3.      By the Realization of Matrix Fraction Descriptions.........................................................................................................	58
	6.6.4.      By the Unilateral Polynomial Matrix Equation................................................................................................................	59
	6.6.5.      By the Equivalence of Block Polynomial Matrices...........................................................................................................	61
	6.6.6.      By Jordan Systems of Polynomial Matrices......................................................................................................................	62
	6.6.7.      By Linear Matrix Equations...............................................................................................................................................	63
	6.6.8.      By Root Functions of Polynomial Matrices......................................................................................................................	65
	6.7.      Sylvester-Polynomial-Conjugate Matrix Equations.......................................................................................................................	66
	6.8.      Generalized Forms of GSE................................................................................................................................................................	70
7.	Iterative Algorithms....................................................................................................................................................................................	75
8.	Applications to GSE....................................................................................................................................................................................	83
	8.1.      Theoretical Applications..................................................................................................................................................................	84
	8.1.1.      Solvability of Matrix Equations.........................................................................................................................................	84
	8.1.2.      UTV Decomposition of Dual Matrices............................................................................................................................	85
	8.1.3.      Microlocal Triangularization of Pseudo-Differential Systems......................................................................................	85
	8.2.      Practical Applications.........................................................................................................................................................................	86
	8.2.1.      Calibration Problems...........................................................................................................................................................	86
	8.2.2.      Encryption and Decryption Schemes for Color Images...................................................................................................	87
9.	Conclusions........................................................................................................................................................................................................	89
10.	References..........................................................................................................................................................................................................	91

1. Introduction

This review commences with the famous Sylvester equation

$$AX-XB=C$$

with unknown X, named after the British mathematician James Joseph Sylvester (1814–1897) [1]. For a detailed discussion of this equation, refer to the review article [2] by Rajendra Bhatia, a Fellow of the Indian National Science Academy. The core of our review revolves around the generalized Sylvester equation (abbreviated as GSE):

$$AX-YB=C$$

with unknown X and Y.

In 1952, Roth established the necessary and sufficient conditions for the solvability of GSE over a field by means of block matrix equivalence. Since then, a large number of papers on GSE have been published in rapid succession. The solvability conditions and explicit expressions of the general solution for GSE have been investigated from diverse perspectives: linear transformations, generalized inverses, matrix decompositions, real (complex) representations, determinant representations, and semi-tensor products, etc. Extensive studies have been conducted on various constrained solutions (e.g., ★-congruent, symmetric, self-adjoint, positive (semi)definite, per(skew)symmetric, bi(skew)symmetric, re-(non)negative definite, re-(non)positive definite, $\eta$-Hermitian, $\eta$-skew-Hermitian, $\varphi$-Hermitian, and equality-constrained solutions) of GSE, as well as its various best approximate solutions when the solvability conditions are not satisfied. Furthermore, the generalizations of GSE in different algebraic structures (e.g., unit regular rings, principal ideal domains, division rings, module-finite rings, commutative rings, Artinian rings, noncommutative rings, dual numbers, dual quaternions), operator equations, tensor equations, polynomial matrix equations, Sylvester-polynomial-conjugate matrix equations, and formal aspects have also shown remarkable vitality. Iterative algorithms for solving various solutions to GSE and its extended forms have also attracted considerable attention, with continuous updates and optimizations. Finally, the relevant results of GSE have demonstrated extraordinary significance in both theoretical applications (e.g., solvability of matrix equations, dual number matrix factorizations, and microlocal triangularization of pseudo-differential systems) and practical applications (e.g., hand-eye calibration problems, and encryption and decryption of color images).

Qing-Wen Wang, the first author of this paper, has been engaged in researching and teaching on the theory of matrix equations since the early 1990s, and has published over 100 related articles. Numerous scholars both domestically and internationally have suggested that we systematically present the theory of solving linear matrix equations. To date, our team has published three review articles, focusing, respectively, on solving the equations $AXB=C$ [3], $AX=C$ and $XB=D$ [4], as well as $A_{1}X{B}_1=C_1$ and $A_{2}X{B}_2=C_2$ [5].

As mentioned earlier, in the development history of research on GSE, its conclusions are interrelated, mutually inspiring, mutually promotive, and progressively advanced, forming a complex, intertwined, and extensive network. By reviewing numerous literature, analyzing and comparing, summarizing, questioning, and prospecting, we strive to identify the commonalities, main threads, and approaches in these studies, aiming to provide insights and inspiration for subsequent research on GSE. Thus, this work is of significant value. In this paper, we intend to focus on GSE as the core theme, and unfold its solution methods, solution categories, generalizations, iterative algorithms, as well as theoretical and practical applications in a step-by-step manner, so as to provide a comprehensive overview.

The remainder of this paper consists of eight sections. Section 2 presents some necessary notations and notes. In Section 3, we introduce Roth’s work on GSE published in [6], which serves as the starting point for this paper. In Section 4, we summarize seven methods for studying GSE, demonstrating diverse perspectives and research schemes. Various solutions to GSE are discussed in detail in Section 5. We devote Section 6 to introducing the generalizations of GSE in certain algebraic aspects. In Section 7, we enumerate several classic iterative algorithms for solving various solutions to GSE and its generalized forms. The theoretical and practical applications of GSE are presented in Section 8. Conclusions are stated in Section 9.

2. Preliminaries

This section recalls some notations and definitions used throughout the paper. Additionally, other necessary terminology for subsequent (sub)sections will be introduced within each. A few symbols, due to their bulk, will be referenced via specific indices in original sources, rather than being reproduced. This not only does not hinder readers’ understanding but also makes the paper more concise and readable.

Let $\mathbb{F}$ be a ring; let $\mathbb{F}\left[\lambda \right]$ be the polynomial ring over $\mathbb{F}$ with the variable $\lambda$; let ${\mathbb{F}}^{m\times n}$ be the set of all $m\times n$ matrices over $\mathbb{F}$; let ${\mathbb{F}}^{m\times n}\left[\lambda \right]$ be the set of all $m\times n$ polynomial matrices over $\mathbb{F}$ with the variable $\lambda$. In particular, ${\mathbb{F}}^m={\mathbb{F}}^{m\times 1}$. Let $degf\left(\lambda \right)$ be the degree of $f\left(\lambda \right)\in \mathbb{F}\left[\lambda \right]$; let $\mathrm{rank}\left(A\right(\lambda \left)\right)$ be the rank of $A\left(\lambda \right)\in {\mathbb{F}}^{m\times n}\left[\lambda \right]$; let $A^T$ be the transpose of $A\in {\mathbb{F}}^{m\times n}$. The symbol $det\left(A\right(\lambda \left)\right)$ denotes the determinant of $A\left(\lambda \right)\in {\mathbb{F}}^{n\times n}\left[\lambda \right]$ for a field $\mathbb{F}$. The component-wise representation of $A\in {\mathbb{F}}^{m\times n}$ can be denoted in the three following forms:

$$A=\left[a_{ij}\right]=\left[a_{i,j}\right]={\left[a_{i,j}\right]}_{m\times n}\in {\mathbb{F}}^{m\times n}.$$

The Kronecker product of $A=\left[a_{ij}\right]\in {\mathbb{F}}^{m\times n}$ and $B\in {\mathbb{F}}^{s\times t}$ is defined by

$$A\otimes B=\left[\begin{array}{cccc}{a}_{11}B& a_{12}B& \dots & a_{1n}B\\ a_{21}B& a_{22}B& \dots & a_{2n}B\\ ⋮& ⋮& ⋮& ⋮\\ a_{m1}B& a_{m2}B& \dots & a_{mn}B\end{array}\right]\in {\mathbb{F}}^{ms\times nt}.$$

The matrices $A\in {\mathbb{F}}^{m\times n}$ and $B\in {\mathbb{F}}^{m\times n}$ are equivalent if there exist two nonsingular matrices $P\in {\mathbb{F}}^{m\times m}$ and $Q\in {\mathbb{F}}^{n\times n}$ such that $A=PBQ$. Moreover, $A\left(\lambda \right)\in {\mathbb{F}}^{m\times n}\left[\lambda \right]$ and $B\left(\lambda \right)\in {\mathbb{F}}^{m\times n}\left[\lambda \right]$ are equivalent if there exist two invertible polynomial matrices $P\left(\lambda \right)\in {\mathbb{F}}^{m\times m}\left[\lambda \right]$ and $Q\left(\lambda \right)\in {\mathbb{F}}^{n\times n}\left[\lambda \right]$ such that $A\left(\lambda \right)=P\left(\lambda \right)B\left(\lambda \right)Q\left(\lambda \right)$.

For a subspace $\mathcal{T}\subseteq {\mathbb{F}}^n$, let $P_{\mathcal{T}}$ be the orthogonal projector onto $\mathcal{T}$; and let ${\mathcal{T}}^{\perp }$ and $dim\left(\mathcal{T}\right)$ be the orthogonal complement and dimension of $\mathcal{T}$, respectively. In addition, denote $A\mathcal{T}=\{At\mid t\in \mathcal{T}\}$ for $A\in {\mathbb{F}}^{m\times n}$. For two vector spaces V and W over a field F, let $\tau$ be a linear transformation from V to W; then $\mathrm{Ker}\left(\tau \right)$ and $\mathrm{Im}\left(\tau \right)$ represent the kernel space and the image of $\tau$, respectively.

Let ${\mathbb{Z}}^{+}$, $\mathbb{N}$, $\mathbb{R}$, and $\mathbb{C}$ be the sets of all positive integers, natural numbers, real numbers, and complex numbers, respectively. The symbol $\mathrm{sign}\left(a\right)$ stands for the sign of $a\in \mathbb{R}$, and ∅ represents the empty set. Let the set of all (real) quaternions be

$$\mathbb{H}=\{q_1+q_2\mathbf{i}+q_3\mathbf{j}+q_4\mathbf{k}\mid {\mathbf{i}}^2={\mathbf{j}}^2={\mathbf{k}}^2=-1,\mathbf{i}\mathbf{j}\mathbf{k}=-1,q_1,q_2,q_3,q_4\in \mathbb{R}\},$$

which is a four-dimensional noncommutative division algebra over $\mathbb{R}$ [7]. Let the conjugate of a quaternion $q=q_1+q_2\mathbf{i}+q_3\mathbf{j}+q_4\mathbf{k}\in \mathbb{H}$ be

$$\overline{q}=q_1-q_2\mathbf{i}-q_3\mathbf{j}-q_4\mathbf{k}.$$

Let $A=\left[a_{ij}\right]\in {\mathbb{H}}^{m\times n}$. The conjugate of A is $\overline{A}=\left[{\overline{a}}_{ij}\right]\in {\mathbb{H}}^{m\times n}$, and the conjugate transpose of A is $A^{*}={\overline{A}}^T$. Let

$$A^{\eta }=-\eta A\eta \mathrm{and}{A^{\eta }}^{*}=-\eta A^{*}\eta ,$$

where $\eta \in \{\mathbf{i},\mathbf{j},\mathbf{k}\}$. If $A={A^{\eta }}^{*}$ ($A=-{A^{\eta }}^{*}$) for $\eta \in \{\mathbf{i},\mathbf{j},\mathbf{k}\}$, then A is called $\eta$-Hermitian ($\eta$-anti-Hermitian) [8]. An $\eta$-anti-Hermitian matrix is also called an $\eta$-skew-Hermitian matrix. Moreover, ${A^{\mathbf{i}}}^{*}=A^{*}$ and ${A^{\mathbf{j}}}^{*}={A^{\mathbf{k}}}^{*}=A^T$ for $A\in {\mathbb{C}}^{m\times n}$.

Denote

$$P_A=A^{†}A,Q_A=A{A}^{†},L_A=I-A^{†}A,R_A=I-A{A}^{†},$$

where $A^{†}$ is the Moore–Penrose inverse [9] of a matrix A. The symbols $A^{-1}$, $\mathrm{rank}\left(A\right)$, $\mathcal{R}\left(A\right)$, and $\mathcal{N}\left(A\right)$ denote the inverse, rank, range, and null space of a matrix A, respectively. In addition, $\overline{A}$, $A^{*}$, and ${\parallel A\parallel }_F$ are the conjugate, conjugate transpose, and Frobenius norm of $A\in {\mathbb{C}}^{m\times n}$, respectively. Let I and 0 be the identity matrix and the null matrix, respectively, with appropriate orders. Specifically, $I_n$ denotes the $n\times n$ identity matrix. For a symmetric matrix $A\in {\mathbb{R}}^{m\times m}$, $A>0$ and $A\ge 0$ denote the positive definite matrix and the positive semidefinite matrix, respectively. In addition, denote $A\ge B$ if $A-B\ge 0$ for matrices A and B. The matrix

$$A=[A_1,A_2,\dots ,A_n]=\left[\begin{array}{cccc}{A}_1& A_2& \dots & A_n\end{array}\right]$$

denotes a new matrix formed by arranging matrix blocks $A_1,A_2,\dots ,A_n$ (with the same number of rows) column-wise. The symbol $\mathrm{diag}({\sigma }_1,\dots ,{\sigma }_r)$ denotes a (block) diagonal matrix with diagonal entries ${\sigma }_1,\dots ,{\sigma }_r$.

For a ring $\mathbb{F}$ and $a\in \mathbb{F}$, a is regular (or inner-invertible) if there exists $a^{-}\in \mathbb{F}$ such that $a{a}^{-}a=a$; a is $(1,2)$-invertible (or reflexive) if there exists $a^{(1,2)}\in \mathbb{F}$ such that $a{a}^{(1,2)}a=a$ and $a^{(1,2)}a{a}^{(1,2)}=a^{(1,2)}$ (see [10]). The characteristic of a field $\mathbb{F}$ is denoted as $\mathrm{char}\left(\mathbb{F}\right)$.

An equation or a system of equations is said to be solvable (consistent) if it has at least one solution. The symbols “⇒” and “⇔” denote “imply” and “if and only if”, respectively. For the reader’s convenience, we provide the main abbreviations used in this paper along with their full names in

Table 1. Abbreviations and their full names.

Full Name	Abbreviation
Generalized Sylvester equation	GSE
Roth’s equivalence theorem	RET
Moore–Penrose inverse	MP inverse
Singular value decomposition	SVD
Generalized singular value decomposition	GSVD
Canonical correlation decomposition	CCD
Semi-tensor product	STP
Second matrix–matrix semi-tensor product	MM-2 STP
Conjugate gradient least-squares algorithm	CGLSA
Alternating direction method	ADM
Relaxed gradient-based iterative algorithm	RGI algorithm

.

At the end of this section, we present three specific notes regarding this review.

(1)

The selection of references follows the core principle of focusing on the theoretical research and practical applications of GSE, with specific criteria as follows:

(i)

Time frame: It covers 19th-century foundational studies to 2025’s latest achievements. It includes classic works like Hamilton’s quaternion research (1844) and Sylvester’s matrix equation study (1884), and emphasizes 2015–2025 recent studies (over 30% of total);

(ii)

Publication venues: Priority is given to peer-reviewed works, including top journals (e.g., SIAM J. Matrix Anal. Appl.), authoritative monographs (e.g., by Gohberg), and key conference papers (e.g., from IEEE ICMA);

(iii)

Content types: Original research is the main focus, with a small number of GSE-related review literature included. Only a few preprints/arXiv works are selected, due to their significance for subsequent research.

(iv)

Relevance scope: Though some research does not focus directly on GSE itself, its traces of relevance to GSE are easily detectable. Thus, we regard such content as an integral part of GSE-related research.

(2)

Results closely related to the theme are rigorously presented as theorems, while less relevant conclusions are briefly summarized narratively. Furthermore, the proofs of these theorems are omitted here.

(3)

The remarks in this paper include comments and suggestions on relevant results, encompassing both previous researchers’ views and our reflections, questions, and prospects.

3. Roth’s Equivalence Theorem

First, we specifically present the paper’s core equation:

$$AX-YB=C,$$

(1)

where $A=\left[a_{i,j}\right]\in {\mathbb{F}}^{m\times r}$, $B=\left[b_{i,j}\right]\in {\mathbb{F}}^{s\times n}$, and $C=\left[c_{i,j}\right]\in {\mathbb{F}}^{m\times n}$ are given over a ring $\mathbb{F}$.

Let $\mathbb{F}$ be a field. In 1952, Roth [6] first studied the necessary and sufficient conditions for the solvability to the polynomial matrix form of Equation (1), i.e.,

$$A\left(\lambda \right)X\left(\lambda \right)-Y\left(\lambda \right)B\left(\lambda \right)=C\left(\lambda \right),$$

(2)

where $A\left(\lambda \right),B\left(\lambda \right),C\left(\lambda \right)\in {\mathbb{F}}^{n\times n}\left[\lambda \right]$, by using the normal form of polynomial matrices.

Theorem 1 

([6]). Equation (2) has a solution pair $X\left(\lambda \right),Y\left(\lambda \right)\in {\mathbb{F}}^{n\times n}\left[\lambda \right]$ if and only if the polynomial matrices

$$\left[\begin{array}{cc}A\left(\lambda \right)& C\left(\lambda \right)\\ 0& B\left(\lambda \right)\end{array}\right]and\left[\begin{array}{cc}A\left(\lambda \right)& 0\\ 0& B\left(\lambda \right)\end{array}\right]$$

are equivalent.

Roth [6] has stated that Theorem 1 remains valid for rectangular polynomial matrices of appropriate orders. Thus, the following theorem can be derived immediately.

Theorem 2 

(Theorem 1, [6]). Let $A\in {\mathbb{F}}^{m\times r}$, $B\in {\mathbb{F}}^{s\times n}$, and $C\in {\mathbb{F}}^{m\times n}$. Then,

$$\mathit{Equation}\left(1\right)\mathit{has}a\mathit{solution}\mathit{pair}X\in {\mathbb{F}}^{r\times n}\mathit{and}Y\in {\mathbb{F}}^{m\times s}$$

(3)

if and only if

$$\left[\begin{array}{cc}A& C\\ 0& B\end{array}\right]\mathit{and}\left[\begin{array}{cc}A& 0\\ 0& B\end{array}\right]\mathit{are}\mathit{equivalent}.$$

(4)

We call this theorem Roth’s equivalence theorem (abbreviated as RET).

Remark 1. 

It is easy to find that the study of Equation (1) is equivalent to that of the equation

$$AX+YB=C$$

(5)

with given $A=\left[a_{i,j}\right]\in {\mathbb{F}}^{m\times r}$, $B=\left[b_{i,j}\right]\in {\mathbb{F}}^{s\times n}$, and $C=\left[c_{i,j}\right]\in {\mathbb{F}}^{m\times n}$, by regarding $-B$ in Equation (1) as B in Equation (5). Thus, in this paper, we collectively refer to Equations (1) and (5) as GSE.

4. Different Methods on GSE

Roth [6] considered Equation (1) based on the canonical forms of the polynomial matrices. Subsequently, many scholars have provided different proofs of RET from various perspectives. These proofs demonstrate the effectiveness and uniqueness of different mathematical methods in considering Equation (1). In addition, these methods have also had a profound impact on the subsequent study of different Sylvester-type equations. This is precisely one of the enchantments of mathematics: reaching the same endpoint through different paths, or even completely unrelated ones. Furthermore, mathematicians ceaselessly pursue groundbreaking innovations and perpetually strive to discover the most elegant path.

Remark 2. 

It is notable that it is easy to see that (3) implies (4) by

$$\left[\begin{array}{cc}I& -Y\\ 0& I\end{array}\right]\left[\begin{array}{cc}A& 0\\ 0& B\end{array}\right]\left[\begin{array}{cc}I& X\\ 0& I\end{array}\right]=\left[\begin{array}{cc}A& AX-YB\\ 0& B\end{array}\right].$$

Therefore, one only needs to consider the sufficiency of RET, i.e., (4) ⇒ (3).

4.1. Method by Linear Transformations and Subspace Dimensions

Let $\mathbb{F}$ be a field. Flanders and Wimmer [11] proved RET for $m=r$ and $s=n$ by means of linear transformations and subspace dimension arguments. This method is more fundamental and elementary than Roth’s method.

Proof of RET [11]. 

Step 1:

Define ${\psi }_i:M_{r+s,2(r+s)}\to M_{r+s}$ by

$$\begin{array}{c}\hfill {\psi }_0(U,W)=\left[\begin{array}{cc}A& 0\\ 0& B\end{array}\right]U-W\left[\begin{array}{cc}A& 0\\ 0& B\end{array}\right]\mathrm{and}{\psi }_1(U,W)=\left[\begin{array}{cc}A& C\\ 0& B\end{array}\right]U-W\left[\begin{array}{cc}A& 0\\ 0& B\end{array}\right].\end{array}$$

Then, the condition (4) yields

$$dim\left(\mathrm{Ker}\left({\psi }_0\right)\right)=dim\left(\mathrm{Ker}\left({\psi }_1\right)\right).$$

Step 2: 

Let

$$U=\left[\begin{array}{cc}{U}_{11}& U_{12}\\ U_{21}& U_{22}\end{array}\right]\mathrm{and}W=\left[\begin{array}{cc}{W}_{11}& W_{12}\\ W_{21}& W_{22}\end{array}\right].$$

Then,

$$\begin{array}{cc}\hfill & \mathrm{Ker}\left({\psi }_0\right)=\left\{(U,W)\left|\begin{array}{cc}A{U}_{11}=W_{11}A,\hfill & A{U}_{12}=W_{12}B,\hfill \\ B{U}_{21}=W_{21}A,\hfill & B{U}_{22}=W_{22}B,\hfill \end{array}\right.\right\},\hfill \\ \hfill & \mathrm{Ker}\left({\psi }_1\right)=\left\{(U,W)\left|\begin{array}{cc}A{U}_{11}+C{U}_{21}=W_{11}A,& A{U}_{12}+C{U}_{22}=W_{12}B,\\ B{U}_{21}=W_{21}A,& B{U}_{22}=W_{22}B,\end{array}\right.\right\}.\hfill \end{array}$$

Let

$$Z=\left\{{\displaystyle [}\begin{array}{cccc}{U}_{21}& U_{22}& W_{21}& W_{22}\end{array}\right]\mid B{U}_{21}=W_{21}A,B{U}_{22}=W_{22}B\}.$$

For $i=0,1$, define

$${\nu }_i:\mathrm{Ker}\left({\psi }_i\right)\to Z,{\nu }_i(U,W)=\left[\begin{array}{cccc}{U}_{21}& U_{22}& W_{21}& W_{22}\end{array}\right].$$

Then, $\mathrm{Im}\left({\nu }_1\right)\subseteq \mathrm{Im}\left({\nu }_0\right)=Z\mathrm{and}\mathrm{Ker}\left({\nu }_1\right)=\mathrm{Ker}\left({\nu }_0\right).$ So, $\mathrm{Im}\left({\nu }_1\right)=\mathrm{Im}\left({\nu }_0\right)$.

Step 3: 

Since $(U,W)\in Ker\left({\psi }_0\right)$ with $U_{22}=-I$, there also exists such in $\mathrm{Ker}\left({\psi }_1\right)$. Therefore, $A{U}_{12}-W_{12}B=C$, i.e., (3) holds.

□

Remark 3. 

(1) 

In [11], Flanders and Wimmer mentioned that, by making small modifications to the above proof, one can similarly obtain the proof of RET under the condition of rectangular matrices A, B, and C.

(2) 

In terms of linear transformations and subspace dimensions, Dmytryshyn et al. discussed two highly complex systems of equations (see Theorem 1.1 in [12] and Theorem 1 in [13]).

4.2. Method by Generalized Inverses

Let $\mathbb{F}$ be a field. In 1955, Penrose [9] concisely defined the Moore–Penrose inverse of any rectangular matrix through four matrix equations.

Definition 1 

([9,10]). Let $A\in {\mathbb{F}}^{m\times n}$. If there exists $X\in {\mathbb{F}}^{n\times m}$ such that

$$\begin{array}{c}\hfill \left(1\right)AXA=A,\left(2\right)XAX=X,\left(3\right){\left(AX\right)}^{*}=AX,\left(4\right){\left(XA\right)}^{*}=XA,\end{array}$$

then X is called the Moore–Penrose (abbreviated as MP) inverse of A and is denoted by $A^{†}$. Especially if $X\in {\mathbb{F}}^{n\times m}$ satisfies the above Equation $\left(1\right)$, then X is called an inner inverse of A and is denoted by $A^{-}$. Clearly, $A^{†}$ is a special inner inverse of A.

Since then, the theory of generalized inverses has flourished (see [10,14,15,16,17,18]), and it has been widely used to study the solvability conditions of linear equations and represent the explicit expressions of their general solutions when solvable. Penrose’s study on the matrix equation $AXB=C$ [9] is the most renowned.

In 1979, Baksalary and Kala [19] naturally utilized inner inverses to establish solvability conditions and representations of the general solution for Equation (1).

Theorem 3 

([19]). Equation (1) has a solution pair $X\in {\mathbb{F}}^{r\times n}$ and $Y\in {\mathbb{F}}^{m\times s}$ if and only if

$$\left(I-A{A}^{-}\right)C\left(I-B^{-}B\right)=0,$$

(6)

in which case

$$\begin{array}{cc}\hfill X& =A^{-}C+A^{-}ZB+\left(I-A^{-}A\right)W,\hfill \\ \hfill Y& =-\left(I-A{A}^{-}\right)C{B}^{-}+Z-\left(I-A{A}^{-}\right)ZB{B}^{-},\hfill \end{array}$$

where W and Z are arbitrary matrices with appropriate orders.

Remark 4. 

In views of RET and Theorem 3, Baksalary and Kala (Remark 2, [19]) noted that

$$\left(6\right)\iff \left(4\right)\iff \mathrm{rank}\left[\begin{array}{cc}A& C\\ 0& B\end{array}\right]=\mathrm{rank}\left(A\right)+\mathrm{rank}\left(B\right).$$

They also pointed out that this result can be directly derived by using (Formula (8.7), [20]) in a particular case, i.e.,

$$\mathrm{rank}\left[\begin{array}{cc}A& C\\ 0& B\end{array}\right]=\mathrm{rank}\left(A\right)+\mathrm{rank}\left(B\right)+\mathrm{rank}\left((I-A{A}^{-})C(I-B^{-}B)\right).$$

Moreover, they said that proof using generalized inverses is simpler than Roth’s [6] and Flanders et al.’s [11]. In fact, this is indeed the case when considering the simplicity of the proof and its length.

Remark 5. 

Under the hypotheses of Theorem 3, it is easy to obtain that

$$\begin{array}{cc}\hfill \left(6\right)& \iff \mathcal{R}\left(C\left(I-B^{-}B\right)\right)\subseteq \mathcal{N}\left(\left(I-A{A}^{-}\right)\right)\hfill \end{array}$$

$$\begin{array}{c}\hfill \iff C\mathcal{N}\left(B\right)\subseteq \mathcal{R}\left(A\right),\end{array}$$

(7)

which is also proved by Woude in (Lemma 3.2, [21]) according to an elementary method. In addition, Woude applied this result to a control problem that occurs in almost non-stationary stochastic processes via measurement feedback (see Theorems 3.3 and 4.1, [21]).

Remark 6. 

It is easy to show that the solvability of Equation (1) is equivalent to the existence of X and Y such that

$$\left[\begin{array}{cc}I& -Y\\ 0& I\end{array}\right]\left[\begin{array}{cc}A& 0\\ 0& B\end{array}\right]\left[\begin{array}{cc}I& X\\ 0& I\end{array}\right]=\left[\begin{array}{cc}A& C\\ 0& B\end{array}\right].$$

(8)

In terms of a geometrical method, Olshevsky gave a cyclic argument over $\mathbb{C}$ as follows:

$$\left(3\right)\Rightarrow \left(8\right)\Rightarrow \left(4\right)\Rightarrow \left(7\right)\Rightarrow \left(3\right),$$

(see the proof of Theorem 1.2 in [22]).

Let $\mathbb{F}=\mathbb{C}$. Meyer [23] revealed an interesting result, that is, the solvability of Equation (1) is equivalent to the existence of the upper block inner inverse of a certain block matrix.

Theorem 4 

(Theorems 1 and 2, [23]). The block triangular complex matrix

$$T=\left[\begin{array}{cc}A& C\\ 0& B\end{array}\right]$$

has an upper block triangular inner inverse if and only if

$$\mathrm{rank}\left(T\right)=\mathrm{rank}\left(A\right)+\mathrm{rank}\left(B\right),$$

in which case,

$$T^{-}=\left[\begin{array}{cc}{A}^{-}& -A^{-}C{B}^{-}\\ 0& B^{-}\end{array}\right],$$

is an inner inverse of T for any $A^{-}$ and $B^{-}$.

4.3. Method by Singular Value Decompositions

Let $\mathbb{F}=\mathbb{R}$. The singular value decomposition (abbreviated as SVD) [24], as an important tool for the study of matrix theory, also plays a significant role in the research on solutions of matrix equations. Chu [25] utilized the SVD to study the solvability conditions and the representation of the general solution of Equation (5).

In fact, let the SVDs of A and B given in Equation (5) be

$$A=U_A{D}_A{V}_A^T\mathrm{and}B=U_B{D}_B{V}_B^T,$$

(9)

where $U_A$, $U_B$, $V_A$, and $V_B$ are orthogonal matrices with appropriate orders,

$$D_A=\left[\begin{array}{cc}{\mathsf{\Sigma }}_A& 0\\ 0& 0\end{array}\right],D_B=\left[\begin{array}{cc}{\mathsf{\Sigma }}_B& 0\\ 0& 0\end{array}\right],$$

${\mathsf{\Sigma }}_A=\mathrm{diag}({\alpha }_1,\dots ,{\alpha }_k)$ with ${\alpha }_i>0$ ($1\le i\le k$), and ${\mathsf{\Sigma }}_B=\mathrm{diag}({\delta }_1,\dots ,{\delta }_l)$ with ${\delta }_j>0$ ($1\le i\le l$). Then,

$$(5)\iff U_A{D}_A{V}_A^{T}X+Y{U}_B{D}_B{V}_B^T=C\iff D_A\tilde{X}+\tilde{Y}{D}_B=\tilde{C},$$

where $\tilde{X}=V_A^{T}X{V}_B$, $\tilde{Y}=U_A^{T}Y{U}_B$, and $\tilde{C}=U_A^{T}C{V}_B$. Partitioning the above equation analogously to the partitioning of $D_A$ and $D_B$, we obtain

$$\begin{array}{cc}\hfill & \left[\begin{array}{cc}{\mathsf{\Sigma }}_A& 0\\ 0& 0\end{array}\right]\left[\begin{array}{cc}{\tilde{X}}_{11}& {\tilde{X}}_{12}\\ {\tilde{X}}_{21}& {\tilde{X}}_{22}\end{array}\right]+\left[\begin{array}{cc}{\tilde{Y}}_{11}& {\tilde{Y}}_{12}\\ {\tilde{Y}}_{21}& {\tilde{Y}}_{22}\end{array}\right]\left[\begin{array}{cc}{\mathsf{\Sigma }}_B& 0\\ 0& 0\end{array}\right]=\left[\begin{array}{cc}{\tilde{C}}_{11}& {\tilde{C}}_{12}\\ {\tilde{C}}_{21}& {\tilde{C}}_{22}\end{array}\right]\hfill \\ \hfill \iff & \left[\begin{array}{cc}{\mathsf{\Sigma }}_A{\tilde{X}}_{11}+{\tilde{Y}}_{11}{\mathsf{\Sigma }}_B& {\mathsf{\Sigma }}_A{\tilde{X}}_{12}\\ {\tilde{Y}}_{21}{\mathsf{\Sigma }}_B& 0\end{array}\right]=\left[\begin{array}{cc}{\tilde{C}}_{11}& {\tilde{C}}_{12}\\ {\tilde{C}}_{21}& {\tilde{C}}_{22}\end{array}\right].\hfill \end{array}$$

(10)

Based on the above discussion, the following theorem can be obtained.

Theorem 5 

(Theorem 2, [25]). Under the notations in (9) and (10), let

$$U_A=\left[\begin{array}{cc}{U}_{A_1}& U_{A_2}\end{array}\right]\mathit{and}{V}_B=\left[\begin{array}{cc}{V}_{B_1}& V_{B_2}\end{array}\right].$$

(1) 

Then, Equation (5) is solvable if and only if

$${\tilde{C}}_{22}=0,i.e.,U_{A_2}^{T}C{V}_{B_2}=0.$$

(11)

(2) 

Suppose that (11) holds. Denote

$$\tilde{C}=\left({\tilde{c}}_{ij}\right)\mathit{and}{M}_{ij}=\left[\begin{array}{cc}{\alpha }_i& {\delta }_j\end{array}\right].$$

Then, ${\tilde{X}}_{21}$, ${\tilde{X}}_{22}$, ${\tilde{Y}}_{12}$, and ${\tilde{Y}}_{22}$ are arbitrary,

$${\tilde{X}}_{12}={\mathsf{\Sigma }}_A^{-1}{\tilde{C}}_{12},{\tilde{Y}}_{21}={\tilde{C}}_{21}{\mathsf{\Sigma }}_B^{-1},{\tilde{X}}_{11}=\left({\tilde{x}}_{ij}\right),\mathrm{and}{\tilde{Y}}_{11}=\left({\tilde{y}}_{ij}\right),$$

where

$$\left[\begin{array}{c}{\tilde{x}}_{ij}\\ {\tilde{y}}_{ij}\end{array}\right]=M_{ij}^{†}{\tilde{c}}_{ij}+(I-M_{ij}^{†}{M}_{ij})Z_{ij}$$

for arbitrary $Z_{ij}$. Moreover, if ${\tilde{Y}}_{11}$ is arbitrary, then

$${\tilde{X}}_{11}={\mathsf{\Sigma }}_A^{-1}({\tilde{C}}_{11}-{\tilde{Y}}_{11}{\mathsf{\Sigma }}_B).$$

(3) 

If (11) holds, then

$$\begin{array}{cc}\hfill X& =V_A\left[\begin{array}{cc}{\mathsf{\Sigma }}_A^{-1}(U_{A_1}^{T}C{V}_{B_1}-Z_1{\mathsf{\Sigma }}_B)& {\mathsf{\Sigma }}_A^{-1}{U}_{A_1}^{T}C{V}_{B_1}\\ Z_2& Z_3\end{array}\right]V_B^T,\hfill \\ \hfill Y& =U_A\left[\begin{array}{cc}{Z}_1& Z_4\\ U_{A_2}^{T}C{V}_{B_1}{\mathsf{\Sigma }}_B^{-1}& Z_5\end{array}\right]U_B^T,\hfill \end{array}$$

where $Z_1,\dots ,Z_5$ are arbitrary matrices with appropriate orders.

Remark 7. 

Interestingly, Chu in Theorem 1 of [25] utilized the generalized singular value decomposition (abbreviated as GSVD) [26] to study the extended form of Equation (5) over $\mathbb{R}$, i.e.,

$$AXE+FYB=C,$$

(12)

where $A,E,F,B$, and C are given real matrices with appropriate orders. Eleven years later, Xu et al. [27] once again discussed the solvability conditions, the general solution, and least-squares solutions of Equation (12) over $\mathbb{C}$ by using the canonical correlation decomposition (abbreviated as CCD) introduced in [28].

Remark 8. 

Inspired by solving Equation (5) via SVD, one can utilize equivalent normal forms of A and B to study Equation (5) over a field, along with an approach similar to Theorem 5. In fact, we only need to regard orthogonal matrices $U_A$, $U_B$, $V_A$, and $V_B$ in (9) as invertible matrices, the transpose $V_A^T$ and $V_B^T$ as the inverse $V_A^{-1}$ and $V_B^{-1}$, and ${\mathsf{\Sigma }}_A$ and ${\mathsf{\Sigma }}_B$ as identity matrices of appropriate orders.

It can be found that using the equivalent normal form of matrices to solve matrix equations is also a very convenient method. This viewpoint has been confirmed multiple times in subsequent research. Wang et al. used the elementary row and column transformations of matrices to give the equivalent normal form of a matrix triplet

$$\left[\begin{array}{ccc}A& B& C\end{array}\right]$$

with the same row number over an arbitrary division ring $\mathbb{F}$ (see Theorem 2.1, [29]). Similarly, the equivalent normal form of another matrix triplet

$${\left[\begin{array}{ccc}{D}^T& E^T& F^T\end{array}\right]}^T$$

with the same column number can also be obtained (see Theorem 2.2, [29]). The two equivalent normal forms are applied to solve the matrix equation

$$AXD+BYE+CZF=G,$$

(13)

where X, Y, and Z are unknown (see Theorem 3.2, [29]).

Interestingly, He et al. utilized Theorem 2.1 of [29] to propose a simultaneous decomposition of seven matrices over $\mathbb{H}$ (see Theorem 2.3, [30]), i.e.,

$$\left[\begin{array}{cccc}G& A& B& C\\ D& & & \\ E& & & \\ F& & \end{array}\right]$$

and discussed Equation (13) once again by this decomposition (see Theorem 3.1, [30]). Compared with the method in [29], which directly applies the equivalent normal forms of two matrix triplets to matrix G, the simultaneous decomposition is more concise. In addition, Ref. [31] once again brilliantly demonstrates the important role of simultaneous decomposition in solving matrix equations.

4.4. Method by Simultaneous Decompositions

Let $\mathbb{F}$ be a field. Remark 8 notes using equivalent canonical forms of the matrices A and B to solve Equation (1). Can we find an equivalent canonical form that is simultaneously related to the matrix triplet

$$\left[\begin{array}{cc}C& A\\ B\end{array}\right]$$

(14)

so as to discuss Equation (1) more simply? Gustafson [32] gave a positive answer to this problem.

Theorem 6 

([32]). Let A, B, and C be given in Equation (1). Then, there exist invertible matrices T, U, V, and W of appropriate orders such that

$$\left[\begin{array}{cc}TCW& TAU\\ VBW\end{array}\right]=\left[\begin{array}{cc}{C}^{\prime }& A^{\prime }\\ B^{\prime }\end{array}\right],$$

(15)

where

$$A^{\prime }=\left[\begin{array}{cccc}{I}_z& 0& 0& 0\\ 0& I_{t_3}& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& I_{r_2}\\ 0& 0& 0& 0\end{array}\right],B^{\prime }=\left[\begin{array}{cccccc}{I}_z& 0& 0& 0& 0& 0\\ 0& I_{t_1}& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0\\ 0& 0& 0& I_{r_1}& 0& 0\end{array}\right],C^{\prime }=\left[\begin{array}{cccccc}{I}_z& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0\\ 0& 0& 0& I_{r_1}& 0& 0\\ 0& 0& 0& 0& I_{r_2}& 0\\ 0& 0& 0& 0& 0& I_{t_2}\end{array}\right],$$

$z+t_3+r_2=\mathrm{rank}\left(A\right)$, $z+t_1+r_1=\mathrm{rank}\left(B\right)$, and $z+t_2+r_1+r_2=\mathrm{rank}\left(C\right)$. We call (15) the simultaneous decomposition of (14).

Remark 9. 

By elementary matrix row and column transformations, Gustafson designed the 12-step algorithm to obtain the simultaneous decomposition of (14) (see (Section 3, [32])).

Applying the simultaneous decomposition of (14) to Equation (1) yields

$$\begin{array}{cc}\hfill \left(1\right)& \iff T^{-1}{A}^{\prime }{U}^{-1}X-Y{V}^{-1}{B}^{\prime }{W}^{-1}=T^{-1}{C}^{\prime }{W}^{-1}\hfill \\ \hfill & \iff A^{\prime }{X}^{\prime }-Y^{\prime }{B}^{\prime }=C^{\prime },\hfill \end{array}$$

(16)

where $X^{\prime }=U^{-1}XW$ and $Y^{\prime }=TY{V}^{-1}$. Thus, $X=U{X}^{\prime }{W}^{-1}$ and $Y=T^{-1}{Y}^{\prime }V$.

Theorem 7 

([32]). Equation (16) is solvable if and only if $t_2=0$, in which case,

$$A^{\prime }=\left[\begin{array}{ccccc}{I}_z& 0& 0& 0& 0\\ 0& I_{t_3}& 0& 0& 0\\ 0& 0& 0& 0& 0\\ 0& 0& 0& I_{r_2}& 0\\ 0& 0& 0& 0& 0\end{array}\right],B^{\prime }=\left[\begin{array}{ccccc}{I}_z& 0& 0& 0& 0\\ 0& I_{t_1}& 0& 0& 0\\ 0& 0& 0& 0& 0\\ 0& 0& 0& I_{r_1}& 0\end{array}\right],C^{\prime }=\left[\begin{array}{ccccc}{I}_z& 0& 0& 0& 0\\ 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0\\ 0& 0& 0& I_{r_1}& 0\\ 0& 0& 0& 0& I_{r_2}\end{array}\right],$$

$$X^{\prime }=\left[\begin{array}{ccccc}{X}_{11}& X_{12}& 0& X_{14}& 0\\ X_{21}& X_{22}& 0& X_{24}& 0\\ X_{31}& X_{32}& X_{33}& X_{35}& X_{35}\\ X_{41}& X_{42}& 0& X_{44}& I_{r_2}\end{array}\right],Y^{\prime }=\left[\begin{array}{cccc}{X}_{11}-I_z& X_{12}& Y_{13}& X_{14}\\ X_{21}& X_{22}-I_{t_1}& Y_{23}& X_{24}\\ 0& 0& Y_{33}& 0\\ 0& 0& Y_{43}& -I_{r_1}\\ X_{41}& X_{42}& Y_{53}& X_{44}\end{array}\right],$$

where the $X_{ij}$ and $Y_{ij}$ are arbitrary matrices with appropriate orders.

Remark 10. 

The quiver theory introduced by Gabriel [33] is an important tool in the representation theory of algebras. Gustafson [32] gave a novel interpretation of the existence of the simultaneous decomposition of (14) from the perspective of the quiver theory. Finally, he gave a necessary and sufficient condition for Equation (1) to have a solution by using the corresponding representation of arrows (see Section 10, [32]). Note that refs. [12,34] also make use of the graph theory to discuss linear matrix equations.

Remark 11. 

Wang et al. in Theorem 2.1 of [35] continued with the idea of simultaneous decomposition and decomposed the following matrices over a division ring $\mathbb{F}$:

$$\left[\begin{array}{ccc}A& B& C\\ D& \end{array}\right],$$

(17)

where A, B, and C are of the same row number, and A and D are of the same column number. So, Theorem 6 is a corollary of Theorem 2.1 of [35] (see Corollary 2.2, [35]). Also, it should be noted that Theorem 2.1 of [29] is indeed a special case of Theorem 2.1 of [35] (see Corollary 2.3, [35]). In addition, Wang et al. applied the simultaneous decomposition of (17) to solving two types of systems of matrix equations. Interestingly, He et al. [36] further refined the simultaneous decomposition presented by [35].

He et al. [37] further considered the simultaneous decompositions of two more general forms over $\mathbb{H}$:

$$\left[\begin{array}{c}A\\ B& C& D\\ & & E\end{array}\right]\mathit{and}\left[\begin{array}{c}A\\ B& C\\ & D\\ & E\end{array}\right],$$

where matrices in the same row (or column) have the same number of rows (or columns), and applied them to solving systems of quaternion matrix equations. Recently, Huo et al. [38] generalized the simultaneous decomposition of (17) to quaternion tensors under the Einstein product.

4.5. Method by Real (Complex) Representations

It is well known that an associative algebra $\mathcal{A}$ with finite dimensions over a field $\mathbb{F}$ is isomorphic to a subalgebra of the algebra ${\mathbb{F}}^{n\times n}$, where n is the dimension of $\mathcal{A}$ over $\mathbb{F}$ (see [39]). We now consider the case of complex numbers, i.e., $\mathbb{F}=\mathbb{C}$.

Let $A=A_0+A_1\mathbf{i}\in {\mathbb{C}}^{m\times r}$, where $A_0,A_1\in {\mathbb{R}}^{m\times r}$, and $\mathbf{i}$ is the imaginary unit such that ${\mathbf{i}}^2=-1$. Define a map

$$\varphi :{\mathbb{C}}^{m\times r}\to {\mathbb{R}}^{2m\times 2r}\mathrm{with}\varphi \left(A\right)=\varphi (A_0+A_1\mathbf{i})=\left[\begin{array}{cc}{A}_0& -A_1\\ A_1& A_0\end{array}\right].$$

We call $\varphi \left(A\right)$ a real representation of the complex matrix A [40]. Then, one can check that $\varphi (·)$ is an isomorphism of the real algebra ${\mathbb{C}}^{m\times r}$ onto the real subalgebra

$$\left\{\left[\begin{array}{cc}{S}_0& -S_1\\ S_1& S_0\end{array}\right]\mid S_0,S_1\in {\mathbb{R}}^{m\times r}\right\}.$$

Then,

$$\left(5\right)\iff \varphi \left(A\right)\varphi \left(X\right)+\varphi \left(Y\right)\varphi \left(B\right)=\varphi \left(C\right).$$

On the other hand, suppose that a real matrix pair

$$\widehat{X}=\left[\begin{array}{cc}{X}_{11}& X_{12}\\ X_{21}& X_{22}\end{array}\right]\mathrm{and}\widehat{Y}=\left[\begin{array}{cc}{Y}_{11}& Y_{12}\\ Y_{21}& Y_{22}\end{array}\right]$$

is a solution pair of the following equation

$$\varphi \left(A\right)\widehat{X}+\widehat{Y}\varphi \left(B\right)=\varphi \left(C\right).$$

(18)

Then, $K_{2r}^{-1}\widehat{X}{K}_{2n}$ and $K_{2m}^{-1}\widehat{Y}{K}_{2s}$ are also a solution pair of Equation (18), where

$$K_{2t}=\left[\begin{array}{cc}0& I_t\\ -I_t& 0\end{array}\right]\mathrm{for}t=m,n,r,s.$$

Let $\overline{X}={\displaystyle \frac{1}{2}}(X_{11}+X_{22})+{\displaystyle \frac{1}{2}}(X_{21}-X_{12})\mathbf{i}$ and $\overline{Y}={\displaystyle \frac{1}{2}}(Y_{11}+Y_{22})+{\displaystyle \frac{1}{2}}(Y_{21}-Y_{12})\mathbf{i}$. Then,

$$\begin{array}{cc}\hfill \varphi \left(\overline{X}\right)& ={\displaystyle \frac{1}{2}}\left[\begin{array}{cc}{X}_{11}+X_{22}& X_{12}-X_{21}\\ X_{21}-X_{12}& X_{11}+X_{22}\end{array}\right]={\displaystyle \frac{1}{2}}\left(\widehat{X}+K_{2r}^{-1}\widehat{X}{K}_{2n}\right),\hfill \\ \hfill \mathrm{and}\varphi \left(\overline{Y}\right)& ={\displaystyle \frac{1}{2}}\left[\begin{array}{cc}{Y}_{11}+Y_{22}& Y_{12}-Y_{21}\\ Y_{21}-Y_{12}& Y_{11}+Y_{22}\end{array}\right]={\displaystyle \frac{1}{2}}\left(\widehat{Y}+K_{2m}^{-1}\widehat{Y}{K}_{2s}\right)\hfill \end{array}$$

satisfy (18), so $\overline{X}$ and $\overline{Y}$ satisfy Equation (5).

Based on the above analysis, the problem of a complex matrix equation is transformed into a problem of a real matrix equation by using the real representation of complex matrices. Based on this consideration, Liu [40] proposed the following theorem for solving Equation (5) over $\mathbb{C}$.

Theorem 8 

(Lemma 1.3, [40]). Let

$$A=A_0+A_1\mathbf{i},B=B_0+B_1\mathbf{i},\mathit{and}C=C_0+C_1\mathbf{i}$$

be given in Equation (5). Then, Equation (5) is consistent over $\mathbb{C}$ if and only if

$$\left[\begin{array}{cc}{A}_0& -A_1\\ A_1& A_0\end{array}\right]\left[\begin{array}{cc}{X}_{11}& X_{12}\\ X_{21}& X_{22}\end{array}\right]+\left[\begin{array}{cc}{Y}_{11}& Y_{12}\\ Y_{21}& Y_{22}\end{array}\right]\left[\begin{array}{cc}{B}_0& -B_1\\ B_1& B_0\end{array}\right]=\left[\begin{array}{cc}{C}_0& -C_1\\ C_1& C_0\end{array}\right]$$

(19)

is consistent over $\mathbb{R}$, in which case,

$$\left\{\begin{array}{c}X=X_0+X_1\mathbf{i}={\displaystyle \frac{1}{2}}(X_{11}+X_{22})+{\displaystyle \frac{1}{2}}(X_{21}-X_{12})\mathbf{i},\hfill \\ Y=Y_0+Y_1\mathbf{i}={\displaystyle \frac{1}{2}}(Y_{11}+Y_{22})+{\displaystyle \frac{1}{2}}(Y_{21}-Y_{12})\mathbf{i},\hfill \end{array}\right.$$

(20)

where $X_{11},X_{12},X_{21},X_{22},Y_{11},Y_{12},Y_{21}$, and $Y_{22}$ constitute the general solution of Equation (19). Furthermore, the explicit forms of X and Y given in (20) are

$$\begin{array}{cc}\hfill X_0=& {\displaystyle \frac{1}{2}}{P}_1\varphi {\left(A\right)}^{-}\varphi \left(C\right)Q_1+{\displaystyle \frac{1}{2}}{P}_2\varphi {\left(A\right)}^{-}\varphi \left(C\right)Q_2+[U_1,U_2]\left[\begin{array}{c}\varphi \left(B\right)Q_1\\ \varphi \left(B\right)Q_2\end{array}\right]+[P_1{F}_{\varphi \left(A\right)},P_2{F}_{\varphi \left(A\right)}]\left[\begin{array}{c}{V}_1\\ V_2\end{array}\right],\hfill \\ \hfill X_1=& {\displaystyle \frac{1}{2}}{P}_2\varphi {\left(A\right)}^{-}\varphi \left(C\right)Q_1-{\displaystyle \frac{1}{2}}{P}_1\varphi {\left(A\right)}^{-}\varphi \left(C\right)Q_2+[U_1,U_2]\left[\begin{array}{c}-\varphi \left(B\right)Q_2\\ \varphi \left(B\right)Q_1\end{array}\right]+[P_2{F}_{\varphi \left(A\right)},-P_1{F}_{\varphi \left(A\right)}]\left[\begin{array}{c}{V}_1\\ V_2\end{array}\right],\hfill \\ \hfill Y_0=& {\displaystyle \frac{1}{2}}{S}_1{E}_{\varphi \left(A\right)}\varphi \left(C\right)\varphi {\left(B\right)}^{-}{T}_1+{\displaystyle \frac{1}{2}}{S}_2{E}_{\varphi \left(A\right)}\varphi \left(C\right)\varphi {\left(B\right)}^{-}{T}_2-[S_1\varphi \left(A\right),S_2\varphi \left(A\right)]\left[\begin{array}{c}{\widehat{U}}_1\\ {\widehat{U}}_2\end{array}\right]+[W_1,W_2]\left[\begin{array}{c}{E}_{\varphi \left(B\right)}{T}_1\\ E_{\varphi \left(B\right)}{T}_2\end{array}\right],\hfill \\ \hfill Y_1=& {\displaystyle \frac{1}{2}}{S}_2{E}_{\varphi \left(A\right)}\varphi \left(C\right)\varphi {\left(B\right)}^{-}{T}_1-{\displaystyle \frac{1}{2}}{S}_1{E}_{\varphi \left(A\right)}\varphi \left(C\right)\varphi {\left(B\right)}^{-}{T}_2+[-S_2\varphi \left(A\right),S_1\varphi \left(A\right)]\left[\begin{array}{c}{\widehat{U}}_1\\ {\widehat{U}}_2\end{array}\right]+[W_1,W_2]\left[\begin{array}{c}{E}_{\varphi \left(B\right)}{T}_1\\ -E_{\varphi \left(B\right)}{T}_2\end{array}\right],\hfill \end{array}$$

where $P_1=[I_r,0],P_2=[0,I_r],S_1=[I_m,0],S_2=[0,I_m]$,

$$\begin{array}{cc}\hfill Q_1& =\left[\begin{array}{c}{I}_n\\ 0\end{array}\right],Q_2=\left[\begin{array}{c}0\\ I_n\end{array}\right],T_1=\left[\begin{array}{c}{I}_s\\ 0\end{array}\right],T_2=\left[\begin{array}{c}0\\ I_s\end{array}\right],\hfill \end{array}$$

and $U_i,V_i,{\widehat{U}}_i,W_i(i=1,2)$ are arbitrary matrices over $\mathbb{R}$ with appropriate orders.

Remark 12. 

Liu [40] further used Theorem 8 to discuss the maximal and minimal ranks of Equation (5)’s solutions, as seen in Section 5.6 of this paper.

Remark 13. 

We know that the quaternion algebra (or the quaternion division ring) cannot be a field because it does not satisfy the commutative law. However, when studying matrix equations over $\mathbb{H}$, the method of converting the quaternion matrix equations to the real (or complex) matrix equations by using the real (or complex) representation of quaternions [7] is widely applied. For example, refs. [41,42] studied the special least-squares solutions of a class of quaternion matrix equations by using the real representation and the complex representation of quaternions, respectively. Moreover, the real (or complex) representation of some other quaternions mentioned in Remark 43 have been explored in [43,44,45,46].

On the other hand, because real (or complex) representations have the drawback of a significant increase in computational load, Wei et al. [47] introduced real structure-preserving methods over $\mathbb{H}$ for LU decomposition, QR decomposition, and SVD, thereby solving quaternion linear systems.

4.6. Method by Determinable Representations

As is well known, Cramer’s rule for the linear equation $Ax=b$ for unknown vector x is an effective means of expressing its unique solution. In addition, it can be found in Section 4.2 that the theory of generalized inverses is closely related to the study of Equation (1). Naturally, we can consider whether Cramer’s rule for the solutions of Equation (1) can be obtained through the determinant representations of generalized inverses.

Generalized inverses (especially the MP inverse) over fields (particularly the complex field) have been thoroughly discussed and applied to characterize various solutions of matrix equations (see [10,15,48]). Notably, Kyrchei [49] presented the determinantal representations of the MP inverse and the Drazin inverse [50] from the new perspective of limit representations of generalized inverses in 2008.

When we consider the determinantal representations of generalized inverses in the quaternion algebra, it relies on the theory of determinants of quaternion matrices. However, due to the noncommutativity of quaternions, the determinants of quaternion matrices become much more complicated (see [51,52]). It was not until several decades later, after Kyrchei [53,54] introduced the theory of column-row determinants over $\mathbb{H}$, that this problem was effectively solved.

Definition 2 

(Column-row determinants over $\mathbb{H}$). (Definitions 2.4 and 2.5, [53]) Let $A=\left[a_{ij}\right]\in {\mathbb{H}}^{n\times n}$, and let $S_n$ be the symmetric group on $I_n=\{1,2,\dots ,n\}$.

(1) 

For $i=1,2,\dots ,n$, the i-th row determinant of A is defined by

$$\begin{array}{cc}\hfill {\mathrm{rdet}}_{i}A& =\sum _{\sigma \in S_n}{(-1)}^{n-r}(a_{i{i}_{k_1}}{a}_{i_{k_1}{i}_{k_1+1}}\dots a_{i_{k_1+l_1}i})\dots (a_{i_{k_r}{i}_{k_r+1}}\dots a_{i_{k_r+l_r}{i}_{k_r}}),\hfill \\ \hfill \sigma & =(i{i}_{k_1}{i}_{k_1+1}\dots i_{k_1+l_1})(i_{k_2}{i}_{k_2+1}\dots i_{k_2+l_2})\dots (i_{k_r}{i}_{k_r+1}\dots i_{k_r+l_r}),\hfill \end{array}$$

where $i_{k_2}<i_{k_3}<\dots <i_{k_r}$, and $i_{k_t}<i_{k_t+s}$ for $t=2,\dots ,r$ and $s=1,\dots ,l_t$.

(2) 

For $j=1,2,\dots ,n$, the j-th column determinant of A is defined by

$$\begin{array}{cc}\hfill {\mathrm{cdet}}_{j}A& =\sum _{\tau \in S_n}{(-1)}^{n-r}(a_{j_{k_r}{j}_{k_r+l_r}}\dots a_{j_{k_r+1}{j}_{k_r}})\dots (a_{j{j}_{k_1+l_1}}\dots a_{j_{k_1+1}{j}_{k_1}}{a}_{j_{k_1}j}),\hfill \\ \hfill \tau & =(j_{k_r+l_r}\dots j_{k_r+1}{j}_{k_r})\dots (j_{k_2+l_2}\dots j_{k_2+1}{j}_{k_2})(j_{k_1+l_1}\dots j_{k_1+1}{j}_{k_1}j),\hfill \end{array}$$

where $j_{k_2}<j_{k_3}<\dots <j_{k_r}$ and $j_{k_t}<j_{k_t+s}$ for $t=2,\dots ,r$ and $s=1,\dots ,l_t$.

Remark 14. 

Kyrchei in Theorem 3.1 of [53] showed that if $A\in {\mathbb{H}}^{n\times n}$ is Hermitian, i.e., $A=A^{*}$, then

$${\mathrm{rdet}}_{1}A=\dots ={\mathrm{rdet}}_{n}A={\mathrm{cdet}}_{1}A=\dots ={\mathrm{cdet}}_{n}A\in \mathbb{R}.$$

Thus, Remark 3.1 [53] defines the determinant of a Hermitian matrix A by

$$detA={\mathrm{rdet}}_{i}A={\mathrm{cdet}}_{i}A,(i=1,2,\dots ,n).$$

We now introduce some necessary notations. Let $A\in {\mathbb{H}}^{m\times n}$. Then, ${\mathcal{R}}_l\left(A\right)$, ${\mathcal{R}}_r\left(A\right)$, ${\mathcal{N}}_l\left(A\right)$, and ${\mathcal{N}}_r\left(A\right)$ denote the left row space, the right column space, the left null space, and the right null space of A, respectively. Let

$$\alpha =\{{\alpha }_1,\dots ,{\alpha }_k\}\subseteq \{1,\dots ,m\}\mathrm{and}\beta =\{{\beta }_1,\dots ,{\beta }_k\}\subseteq \{1,\dots ,n\},$$

where $1\le k\le min\{m,n\}$. By $A_{\beta }^{\alpha }$, we denote the submatrix of A whose rows are indexed by $\alpha$ and whose columns are indexed by $\beta$. If A is Hermitian, then $|A_{\alpha }^{\alpha }|$ is the corresponding principal minor of A. For $1\le k\le n$, let

$$L_{k,n}=\{\alpha \mid \alpha =({\alpha }_1,\dots ,{\alpha }_k),1\le {\alpha }_1<\dots <{\alpha }_k\le n\}.$$

And, for $i\in \alpha$ and $j\in \beta$, let

$$I_{r,m}\left\{i\right\}=\{\alpha \mid \alpha \in L_{r,m},i\in \alpha \}\mathrm{and}{J}_{r,n}\left\{j\right\}=\{\beta \mid \beta \in L_{r,n},j\in \beta \}.$$

We denote the j-th column of A by ${\mathbf{a}}_{.j}$ and its i-th row by ${\mathbf{a}}_{i.}$. And, $A_{.j}\left(\mathbf{b}\right)$ is the matrix that is obtained from A by replacing its j-th column by the column vector $\mathbf{b}\in {\mathbb{H}}^{m\times 1}$, and $A_{i.}\left(\mathbf{b}\right)$ is the matrix obtained from A by replacing its i-th row by the row vector $\mathbf{b}\in {\mathbb{H}}^{1\times n}$. Symbols ${\mathbf{a}}_{·j}^{*}$ and ${\mathbf{a}}_{i·}^{*}$ denoted the j-th column and the i-th row of $A^{*}$, respectively.

Kyrchei [55] obtained the Cramer’s rules for some left, right and two-sided quaternion matrix equations by the theory of the column-row determinants over $\mathbb{H}$. Song et al. [56] then utilized results in [55] to further study the Cramer’s rule for the quaternion matrix equation:

$$AXE+FYB=C,$$

(21)

where $A\in {\mathbb{H}}^{m\times n}$, $E\in {\mathbb{H}}^{s\times q}$, $F\in {\mathbb{H}}^{m\times q}$, $B\in {\mathbb{H}}^{t\times q}$, and $C\in {\mathbb{H}}^{m\times q}$ are given.

Theorem 9 

(Theorem 3.1, [56]). Suppose that Equation (21) is consistent. Let

$$\begin{array}{cc}\hfill & T=A^{*}(I+R_F)A,S=E(I+L_B)E^{*},A_{11}=A^{*}{R}_{F}A{T}^{†},A_{22}=A^{*}A{T}^{†},\hfill \\ \hfill & E_{11}=S^{†}E{E}^{*},E_{22}=S^{†}E{L}_B{E}^{*},Y_{10}=A_{22}^{†}{A}_{11}{A}^{*}C{L}_B{E}^{*}+L_{A_{22}}{A}^{*}{R}_{F}C{E}^{*}{E}_{22}{E}_{11}^{†},\hfill \\ \hfill & Y_{20}=A_{11}^{†}{A}_{22}{A}^{*}{R}_{F}C{E}^{*}+L_{A_{11}}{A}^{*}C{L}_B{E}^{*}{E}_{11}{E}_{22}^{†}.\hfill \end{array}$$

Let $K^{*}$, L, $M^{*}$, and N be of full column rank matrices over $\mathbb{H}$ such that

$${\mathcal{N}}_r\left(T\right)={\mathcal{R}}_r\left(K^{*}\right),{\mathcal{N}}_r\left(S\right)={\mathcal{R}}_r\left(L\right),{\mathcal{N}}_r\left(F\right)={\mathcal{R}}_r\left(M^{*}\right),{\mathcal{N}}_r\left(B\right)={\mathcal{R}}_r\left(N\right).$$

Then, the general solution of Equation (21) is

$$\begin{array}{cc}\hfill x_{ij}& ={\displaystyle \frac{{\mathrm{rdet}}_j{(S+L{L}^{*})}_{j.}\left(c_{i.}^A\right)}{\det (T+K^{*}K)\det (S+L{L}^{*})}}={\displaystyle \frac{{\mathrm{cdet}}_i{(T+K^{*}K)}_{.i}\left(c_{.j}^E\right)}{\det (T+K^{*}K)\det (S+L{L}^{*})}},\hfill \\ \hfill y_{hl}& ={\displaystyle \frac{{\mathrm{rdet}}_l{(B{B}^{*}+N{N}^{*})}_{l.}\left(g_{h.}^A\right)}{\det (F^{*}F+M^{*}M)\det (B{B}^{*}+N{N}^{*})}}={\displaystyle \frac{{\mathrm{cdet}}_h{(F^{*}F+M^{*}M)}_{.h}\left(g_{.l}^E\right)}{\det (F^{*}F+M^{*}M)\det (B{B}^{*}+N{N}^{*})}},\hfill \end{array}$$

where

$$\begin{array}{cc}\hfill c_{i.}^A& =\left[{\mathrm{cdet}}_i{(T+K^{*}K)}_{.i}\left(d_{.1}\right),\dots ,{\mathrm{cdet}}_i{(T+K^{*}K)}_{.i}\left(d_{.s}\right)\right],\hfill \\ \hfill g_{h.}^A& =\left[{\mathrm{cdet}}_h{(F^{*}F+M^{*}M)}_{.h}\left(k_{.1}\right),\dots ,{\mathrm{cdet}}_h{(F^{*}F+M^{*}M)}_{.h}\left(k_{.t}\right)\right],\hfill \\ \hfill c_{.j}^E& ={\left[{\mathrm{rdet}}_j{(S+L{L}^{*})}_{j.}\left(d_{1.}\right),\dots ,{\mathrm{rdet}}_j{(S+L{L}^{*})}_{j.}\left(d_{n.}\right)\right]}^T,\hfill \\ \hfill g_{.l}^E& ={\left[{\mathrm{rdet}}_l{(B{B}^{*}+N{N}^{*})}_{l.}\left(k_{1.}\right),\dots ,{\mathrm{rdet}}_l{(B{B}^{*}+N{N}^{*})}_{l.}\left(k_{q.}\right)\right]}^T\hfill \end{array}$$

with $d_{i.}$ and $d_{.j}$ are the i-th row and j-th column of

$$(T+K^{*}K)T^{†}(A^{*}{R}_{F}C{E}^{*}+A^{*}C{L}_B{E}^{*}+Y_{10}+Y_{20})S^{†}(S+L{L}^{*})+M+W,$$

respectively, and $k_{.i}$ and $k_{j.}$ are the i-th column and j-th row of

$$(F^{*}F+M^{*}M)\left(F^{†}(C-AXE)B^{†}\right)(B{B}^{*}+N{N}^{*})+Q,$$

respectively, for $i=1,\dots ,m$, $j=1,\dots ,s$, $h=1,\dots ,q$, and $l=1,\dots ,t$, where

$$\begin{array}{cc}\hfill M& =(T+K^{*}K)T^{†}(L_{A_{22}}{V}_1{R}_{E_{11}}+L_{A_{11}}{V}_2{R}_{E_{22}})S^{†}(S+L{L}^{*}),\hfill \\ \hfill W& =TZL{L}^{*}+K^{*}KZS+K^{*}KZL{L}^{*},Q=F^{*}FHN{N}^{*}+M^{*}MH(B{B}^{*}+N{N}^{*})\hfill \end{array}$$

for arbitrary $V_1$, $V_2$, Z, and H with appropriate orders.

Remark 15. 

When E and F in Equation (21) are identity matrices, by Theorem 9, we immediately derived the Cramer’s rule to the matrix equation over $\mathbb{H}$:

$$AX+YB=C.$$

(22)

Note that Theorem 9 uses the auxiliary matrices, i.e., K, L, M, and N, to derive the determinant representation of the general solution of Equation (22). However, these auxiliary matrices are not always easy to obtain in practical applications. Then, can we obtain Cramer’s rule for Equation (22) only through its given coefficient matrices?

In order to answer the above problem, let us first take a look at another work of Kyrchei [57]. Based on the column-row determinant theory and the limit representation of the MP inverse, he gave the determinant representation of the MP inverse over $\mathbb{H}$ in [57]. This research has greatly promoted the study of Cramer’s rules to the various solutions of matrix equations over $\mathbb{H}$ (see [58,59,60,61,62,63,64]). Denote

$${\mathbb{H}}_r^{m\times n}=\{A\in {\mathbb{H}}^{m\times n}|\mathrm{rank}\left(A\right)=r\}.$$

Theorem 10 

(Theorem 5, [57]). Let $A\in {\mathbb{H}}_r^{m\times n}$ and $A^{†}=\left[a_{i,j}^{+}\right]\in {\mathbb{H}}^{n\times m}$. Then,

$$a_{i,j}^{+}={\displaystyle \frac{{\sum }_{\beta \in J_{r,n}\left\{i\right\}}{\mathrm{cdet}}_i{\left({\left(A^{*}A\right)}_{·i}\left({\mathbf{a}}_{·j}^{*}\right)\right)}_{\beta }^{\beta }}{{\sum }_{\beta \in J_{r,n}}\left|{\left(A^{*}A\right)}_{\beta }^{\beta }\right|}}={\displaystyle \frac{{\sum }_{\alpha \in I_{r,m}\left\{j\right\}}{\mathrm{rdet}}_j{\left({\left(A{A}^{*}\right)}_{j·}\left({\mathbf{a}}_{i·}^{*}\right)\right)}_{\alpha }^{\alpha }}{{\sum }_{\alpha \in I_{r,m}}\left|{\left(A{A}^{*}\right)}_{\alpha }^{\alpha }\right|}},$$

where $i=1,2.\dots ,n$ and $j=1,2,\dots ,m$.

A result similar to Theorem 3 was given by Kyrchei in [58].

Theorem 11 

(Lemma 5.1, [58]). The following are equivalent:

(1) 

Equation (22) is solvable;

(2) 

$R_{A}C{L}_B=0$;

(3) 

$\mathrm{rank}\left[\begin{array}{cc}A& C\\ 0& B\end{array}\right]=\mathrm{rank}\left[\begin{array}{cc}A& 0\\ 0& B\end{array}\right]$;

in which case,

$$\begin{array}{c}\hfill X=A^{†}C-A^{†}VB+L_{A}U\mathit{and}Y={\mathrm{R}}_{A}C{B}^{†}+A{A}^{†}V+W{\mathrm{R}}_B,\end{array}$$

(23)

where U, V, and W are arbitrary matrices of appropriate orders over $\mathbb{H}$.

It is easy to see that

$$\begin{array}{c}\hfill X=A^{†}C\mathrm{and}Y=C{B}^{†}-A{A}^{†}C{B}^{†}\end{array}$$

(24)

are the partial solution pair of Equation (22) by taking U, V, and W as zero matrices in (23). Then, by applying Theorem 10 to (24), Kyrchei [58] presented a new Cramer’s rule for the partial solution of Equation (22), which only makes use of the coefficient matrices, i.e., A, B, and C.

Theorem 12 

(Theorem 5.2, [58]). Let $A\in {\mathbb{H}}_{r_1}^{m\times n}$ and $B\in {\mathbb{H}}_{r_2}^{t\times q}$. Then, the partial solution pair $X=\left[x_{ij}\right]\in {\mathbb{H}}^{n\times q}$ and $Y=\left[y_{gf}\right]\in {\mathbb{H}}^{m\times t}$ in (24) can be expressed as

$$x_{ij}={\displaystyle \frac{{\sum }_{\beta \in J_{r_1,n}\left\{i\right\}}{\mathrm{cdet}}_i{\left({\left(A^{*}A\right)}_{.i}\left({\mathbf{c}}_{.j}^{\left(1\right)}\right)\right)}_{\beta }^{\beta }}{{\sum }_{\beta \in J_{r_1,n}}\left|{\left(A^{*}A\right)}_{\beta }^{\beta }\right|}},$$

where ${\mathbf{c}}_{.j}^{\left(1\right)}$ is the j-th columns of $A^{*}C$, and

$$\begin{array}{cc}\hfill y_{gf}& ={\displaystyle \frac{{\sum }_{\alpha \in I_{r_2,t}\left\{f\right\}}{\mathrm{rdet}}_f{\left({\left(B{B}^{*}\right)}_{.f}\left({\mathbf{c}}_{g.}^{\left(2\right)}\right)\right)}_{\alpha }^{\alpha }}{{\sum }_{\alpha \in I_{r_2,t}}\left|{\left(B{B}^{*}\right)}_{\alpha }^{\alpha }\right|}}\hfill \\ \hfill & -{\displaystyle \frac{{\sum }_{l=1}^m{\sum }_{\alpha \in I_{r_1,m}\left\{l\right\}}{\mathrm{rdet}}_l{\left({\left(A{A}^{*}\right)}_{l.}\left({\ddot{\mathbf{a}}}_{g.}^{\left(1\right)}\right)\right)}_{\alpha }^{\alpha }{\sum }_{\alpha \in I_{r_2,t}\left\{f\right\}}{\mathrm{rdet}}_f{\left({\left(B{B}^{*}\right)}_{.f}\left({\mathbf{c}}_{l.}^{\left(2\right)}\right)\right)}_{\alpha }^{\alpha }}{{\sum }_{\alpha \in I_{r_1,m}}\left|{\left(A{A}^{*}\right)}_{\alpha }^{\alpha }\right|{\sum }_{\alpha \in I_{r_2,t}}\left|{\left(B{B}^{*}\right)}_{\alpha }^{\alpha }\right|}},\hfill \end{array}$$

where ${\mathbf{c}}_{g.}^{\left(2\right)}$ and ${\ddot{\mathbf{a}}}_{g.}^{\left(1\right)}$ are the g-th rows of $C{B}^{*}$ and $A{A}^{*}$, respectively.

Remark 16. 

Note that although Kyrchei [58] gave the determinant representation of the particular solution of Equation (22), the problem proposed in Remark 15 is not solved completely. That is to say, Cramer’s rule for representing the general solution of Equation (22) only using the coefficient matrices remains an unsolved problem.

Interestingly, Song [65] considered the determinant representation for the general solution of Equation (22) with the restricted conditions, i.e.,

$$\mathrm{Equation}(22)\mathrm{subject}\mathrm{to}\left\{\begin{array}{c}{\mathcal{R}}_r\left(X\right)\subseteq T_1,{\mathcal{N}}_r\left(X\right)\supseteq S_1,\hfill \\ {\mathcal{R}}_l\left(Y\right)\subseteq T_2,{\mathcal{N}}_l\left(Y\right)\supseteq S_2,\hfill \end{array}\right.$$

(25)

where $T_1\subseteq {\mathbb{H}}^n$, $S_1\subseteq {\mathbb{H}}^p$, $T_2\subseteq {\mathbb{H}}^{1\times t}$, and $S_2\subseteq {\mathbb{H}}^{1\times m}$. Let $P_{\mathcal{T}}\in {\mathbb{H}}^{n\times n}$ ($Q_{\mathcal{T}}\in {\mathbb{H}}^{n\times n}$) denote the right (left) $\mathbb{H}$-orthogonal projector onto a right (left) $\mathbb{H}$-vector subspace $\mathcal{T}\in {\mathbb{H}}^{n\times 1}$ ($\mathcal{T}\in {\mathbb{H}}^{1\times n}$) along ${\mathcal{T}}^{\perp }$.

Theorem 13 

(Theorem 3.2, [65]). Let A, B, C, $T_1$, $T_2$, $S_1$, and $S_2$ be given in (25), and let

$$M=R_{A{P}_{T_1}}{Q}_{S_2^{\perp }},N=Q_{T_2}B{P}_{S_1^{\perp }},T=A^{*}{R}_{Q_{S_2^{\perp }}}A+A^{*}A,S=I+L_{Q_{T_2}B},$$

and $Y_1$ be such that

$$A^{*}{R}_{Q_{S_2^{\perp }}}A{T}^{†}{Y}_1=A^{*}A{T}^{†}{A}^{*}{R}_{Q_{S_2^{\perp }}}C\mathit{and}{Y}_1{L}_{Q_{T_2}B}=A^{*}C{L}_{Q_{T_2}B}.$$

(1) 

The restricted Equation (25) is solvable if and only if

$$\begin{array}{c}\hfill R_M{R}_{A{P}_{T_1}}C=0,R_{A{P}_{T_1}}C{L}_{Q_{T_2}B}=0,C{L}_{P_{S_1^{\perp }}}{L}_N=0,R_{Q_{S_2^{\perp }}}C{L}_{Q_{T_2}B}=0,\end{array}$$

in which case,

$$\begin{array}{cc}\hfill X& ={\left(T{P}_{T_1}\right)}^{†}\tilde{C}{P}_{S_1^{\perp }}+P_{T_1}{L}_{A{P}_{T_1}}{U}_1{P}_{S_1^{\perp }}+P_{T_1}{V}_1{P}_{S_1^{\perp }}\hfill \\ \hfill & ={\left(T{P}_{T_1}\right)}^{†}\tilde{C}{P}_{S_1^{\perp }}+P_{T_1}{Z}_1{P}_{S_1^{\perp }}-P_{T_{11}}{Z}_1{P}_{S_1^{\perp }}\hfill \\ \hfill & ={\left(T{P}_{T_1}\right)}^{†}\tilde{C}{P}_{S_1^{\perp }}+P_{T_1\cap {\mathcal{N}}_r\left(A\right)}{U}_2{P}_{S_1^{\perp }}+P_{T_1}{V}_2{P}_{S_1^{\perp }},\hfill \\ \hfill Y& =Q_{S_2^{\perp }}(C-AX){\left(Q_{T_2}B\right)}^{†}+Q_{S_2^{\perp }}{V}_3{R}_{Q_{T_2}B}{Q}_{T_2},\hfill \end{array}$$

where

$$\tilde{C}=\left(A^{*}{R}_{Q_{S_2^{\perp }}}C+A^{*}C{L}_{Q_{T_2}B}+A^{*}{R}_{Q_{S_2^{\perp }}}C{L}_{Q_{T_2}B}+Y_1\right)S^{-1},$$

and $Z_1$, $V_1$, $U_1$, $V_2$, $U_2$, and $V_3$ are arbitrary matrices over $\mathbb{H}$ with appropriate dimensions.

(2) 

Let $C_1^{*},K_1^{*},C_1$, and $K_2$ be full column rank matrices such that

$$T_1={\mathcal{N}}_r\left(C_1\right),T_1\cap {\mathcal{N}}_r\left(T\right)={\mathcal{R}}_r\left(K_1^{*}\right),T_2={\mathcal{N}}_l\left(C_2\right),T_2\cap {\mathcal{N}}_r\left(B\right)={\mathcal{R}}_l\left(K_2^{*}\right).$$

Denote $X=\left[x_{ij}\right]\in {\mathbb{H}}^{n\times q}$ and $Y=\left[y_{kl}\right]\in {\mathbb{H}}^{m\times t}$. If Equation (25) is solvable, then

$$\begin{array}{cc}\hfill x_{ij}& ={\displaystyle \frac{{\mathrm{cdet}}_i{(A^{*}A+C_1^{*}{C}_1+K_1^{*}{K}_1)}_{.i}\left(d_{.j}\right)}{det(A^{*}A+C_1^{*}{C}_1+K_1^{*}{K}_1)}},\hfill \\ \hfill y_{kl}& ={\displaystyle \frac{{\mathrm{rdet}}_l{(B{B}^{*}+C_2{C}_2^{*}+K_2{K}_2^{*})}_{l.}\left(d_{k.}\right)}{det(B{B}^{*}+C_2{C}_2^{*}+K_2{K}_2^{*})}}\hfill \end{array}$$

with $d_{.j}$ is the j-th column of

$$T\left(A^{*}{R}_{Q_{S_2^{\perp }}}C+A^{*}C{L}_{Q_{T_2}B}+A^{*}{R}_{Q_{S_2^{\perp }}}C{L}_{Q_{T_2}B}+Y_1\right)+K_1^{*}{K}_1{Z}_1,$$

and $d_{k.}$ is the i-th row of

$$(C-AX)B^{*}+Z_2{K}_2{K}_2^{*},$$

where $i=1,\dots ,n$, $j=1,\dots ,q$, $k=1,\dots ,m$, $l=1,\dots ,t$, and $Z_1$ and $Z_2$ are arbitrary matrices over $\mathbb{H}$ with appropriate orders.

4.7. Method by Semi-Tensor Products

To effectively handle multidimensional arrays and nonlinear problems, Chinese mathematician Daizhan Cheng [66] pioneered a new matrix product in 2001: the semi-tensor product (abbreviated as STP). It exactly coincides with traditional matrix products when the two factor matrices meet the dimension requirements. Due to the excellent properties of STP [67], namely

(1)

It is applied to any two matrices;

(2)

It has certain commutative properties;

(3)

It inherits all properties of the conventional matrix product;

(4)

It enables easy expression of multilinear functions (mappings);

STP has been effectively applied to Boolean (control) networks, logical dynamic systems, system biology, graph theory, formation control, finite automata and symbolic dynamics, circuit design and failure detection, coding and cryptography, fuzzy control, engineering, game theory, and so on (see [67,68,69,70,71] and references therein).

Definition 3 

([66]). Let $\mathbb{F}$ be a ring, $A\in {\mathbb{F}}^{m\times n}$, and $B\in {\mathbb{F}}^{p\times q}$. The left STP of A and B is defined as

$$A⋉B=\left(A\otimes I_{{\displaystyle \frac{t}{n}}}\right)\left(B\otimes I_{{\displaystyle \frac{t}{p}}}\right),$$

where $t=\mathrm{lcm}(n,p)$ is the least common multiple of n and p.

Remark 17. 

Similar to the left STP, Cheng [68] defined the right STP of $A\in {\mathbb{F}}^{m\times n}$ and $B\in {\mathbb{F}}^{p\times q}$, i.e.,

$$A⋊B=\left(I_{{\displaystyle \frac{t}{n}}}\otimes A\right)\left(I_{{\displaystyle \frac{t}{p}}}\otimes B\right),$$

where $t=\mathrm{lcm}(n,p)$. Both (Proposition 1.3, [67]) and (Proposition 2.3.2, [70]) show that the block multiplication of matrices can be extended to the left STP. Unfortunately, the right STP does not satisfy the block multiplication rule, which is a significant difference between the left STP and right STP. Next, we use a simple example from [70] to demonstrate this limitation of the right STP: let $A=\left[a_1{a}_2{a}_3{a}_4\right]$ and $B={\left[b_1{b}_2\right]}^T$. Then,

$$A⋉B=[a_1{b}_1+a_3{b}_2{a}_2{b}_1+a_4{b}_2]\mathit{and}A⋊B=[a_1{b}_1+a_2{b}_2{a}_3{b}_1+a_4{b}_2].$$

Partition A and B equally into $A=\left[A_1{A}_2\right]$ and $B={\left[B_1{B}_2\right]}^T$, respectively. According to the block multiplication rule, we have

$$\begin{array}{c}\hfill A_1⋉B_1+A_2⋉B_2=\left[a_1{a}_2\right]⋉b_1+\left[a_3{a}_4\right]⋉b_2=[a_1{b}_1+a_3{b}_2{a}_2{b}_1+a_4{b}_2],\\ \hfill A_1⋊B_1+A_2⋊B_2=\left[a_1{a}_2\right]⋊b_1+\left[a_3{a}_4\right]⋊b_2=[a_1{b}_1+a_3{b}_2{a}_2{b}_1+a_4{b}_2].\end{array}$$

Obviously,

$$A⋉B=A_1⋉B_1+A_2⋉B_2,\mathit{but}A⋊B\ne A_1⋊B_1+A_2⋊B_2.$$

This drawback makes the right STP unable to replace the left STP in many applications, greatly limiting its use. Therefore, most scholars focus exclusively on research into the left STP. Based on this, the STP referred to in this paper specifically denotes the left STP.

Remark 18. 

As a generalization of the traditional matrix product, and given its extensive application significance, the study of matrix equations under STP has emerged as an inevitable and pivotal research direction. In 2016, Yao et al. [72] were the first to conduct research on the classical complex matrix equation under STP:

$$A⋉X=B,$$

where X is unknown. Subsequently, building on the research framework of [72], investigations into diverse matrix equations under STP have flourished (see [73,74,75,76,77,78]).

Regrettably, no studies directly addressing Equation (5) under STP have been published to date. However, when the dimension of Y is constrained to equal that of $X^T$, the exploration of Sylvester-transpose matrix Equation [74]

$$A⋉X+X^T⋉B=C$$

with unknown X may provide valuable guidance for this problem.

Notably, in 2019, Cheng and Liu [79], inspired by the research on cross-dimensional linear systems [80], further proposed the second matrix–matrix semi-tensor product (abbreviated as MM-2 STP), denoted by ${\circ }_l$. Subsequently, Wang [81] investigated the complex matrix equation under the MM-2 STP:

$$A{\circ }_{l}X=B,$$

where X is unknown, and A and B are given. So, exploring Equation (1) under MM-2 STP emerges as a potential research topic. Additionally, Cheng [67] introduced the dimension-free matrix theory, which systematically elucidates the deep mathematical insights underlying STP.

Since 2020, Liaocheng University’s team led by Professors Ying Li and Jianli Zhao has utilized STP to propose several novel matrix representations, including complex, quaternion, and octonion matrices. These representations have also been applied to solving corresponding matrix equations, and have yielded effective numerical experimental results.

Ding et al. [82] first proposed the real vector representation for quaternion matrices, whose properties were characterized by STP, as follows:

Definition 4 

(Definitions 3.1–3.3, [82]). Let $x=x_1+x_2\mathbf{i}+x_3\mathbf{j}+x_4\mathbf{k}\in \mathbb{H}$. Denote

$$v^R\left(x\right)={\left[\begin{array}{cccc}{x}_1& x_2& x_3& x_4\end{array}\right]}^T.$$

Let $x=\left[\begin{array}{ccc}{x}^1& \dots & x^n\end{array}\right]\in {\mathbb{H}}^{1\times n}$ and $y={\left[\begin{array}{ccc}{y}^1& \dots & y^n\end{array}\right]}^T\in {\mathbb{H}}^n$. Denote

$$v^R\left(x\right)=\left[\begin{array}{c}{v}^R\left(x^1\right)\\ ⋮\\ v^R\left(x^n\right)\end{array}\right]\mathit{and}{v}^R\left(y\right)=\left[\begin{array}{c}{v}^R\left(y^1\right)\\ ⋮\\ v^R\left(y^n\right)\end{array}\right].$$

For $A\in {\mathbb{H}}^{m\times n}$, the real column stacking form $v_c^R\left(A\right)$ and the real row stacking form $v_r^R\left(A\right)$ of A are defined as

$$v_c^R\left(A\right)=\left[\begin{array}{c}{v}^R\left({\mathrm{Col}}_1\left(A\right)\right)\\ v^R\left({\mathrm{Col}}_2\left(A\right)\right)\\ ⋮\\ v^R\left({\mathrm{Col}}_n\left(A\right)\right)\end{array}\right]\mathit{and}{v}_r^R\left(A\right)=\left[\begin{array}{c}{v}^R\left({\mathrm{Row}}_1\left(A\right)\right)\\ v^R\left({\mathrm{Row}}_2\left(A\right)\right)\\ ⋮\\ v^R\left({\mathrm{Row}}_m\left(A\right)\right)\end{array}\right],$$

where ${\mathrm{Col}}_j\left(A\right)(j=1,\dots ,n)$ and ${\mathrm{Row}}_i\left(A\right)(j=1,\dots ,m)$ are the j-th column and i-th row of A, respectively.

Remark 19. 

For $A\in {\mathbb{H}}^{m\times n}$ and $B\in {\mathbb{H}}^{n\times p}$, Theorem 3.3(3) of [82] shows

$$v_r^R\left(AB\right)=G\left(v_r^R\left(A\right)⋉v_c^R\left(B\right)\right),$$

where

$$\begin{array}{cc}\hfill G& =\left[\begin{array}{c}F⋉{\left({\delta }_m^1\right)}^T⋉\left[I_{4mn}\otimes {\left({\delta }_p^1\right)}^T\right]\\ ⋮\\ F⋉{\left({\delta }_m^1\right)}^T⋉\left[I_{4mn}\otimes {\left({\delta }_p^p\right)}^T\right]\\ ⋮\\ F⋉{\left({\delta }_m^m\right)}^T⋉\left[I_{4mn}\otimes {\left({\delta }_p^1\right)}^T\right]\\ ⋮\\ F⋉{\left({\delta }_m^m\right)}^T⋉\left[I_{4mn}\otimes {\left({\delta }_p^p\right)}^T\right]\end{array}\right],F=M_Q⋉\left(\sum _{i=1}^n{\left({\delta }_n^i\right)}^T⋉(I_{4n}\otimes {\left({\delta }_n^i\right)}^T)\right),\hfill \end{array}$$

$$\begin{array}{cc}\hfill M_Q& =\left[\begin{array}{cccccccccccccccc}1& 0& 0& 0& 0& -1& 0& 0& 0& 0& -1& 0& 0& 0& 0& -1\\ 0& 1& 0& 0& 1& 0& 0& 0& 0& 0& 0& 1& 0& 0& -1& 0\\ 0& 0& 1& 0& 0& 0& 0& -1& 1& 0& 0& 0& 0& 1& 0& 0\\ 0& 0& 0& 1& 0& 0& 1& 0& 0& -1& 0& 0& 1& 0& 0& 0\end{array}\right],\hfill \end{array}$$

and ${\delta }_m^i(i=1,\dots ,m)$ is the i-th column of $I_m$.

Using the real vector representation for quaternion matrices, Ding et al. [82] further discussed the special least-squares solutions of quaternion matrix equation

$$AXB+CYD=E,$$

(26)

where $A\in {\mathbb{H}}^{m\times n}$, $B\in {\mathbb{H}}^{n\times s}$, $C\in {\mathbb{H}}^{m\times k}$, $D\in {\mathbb{H}}^{k\times s}$, and $E\in {\mathbb{H}}^{m\times s}$. Denote

$${\delta }_k[i_1,\dots ,i_s]=[{\delta }_k^{i_1},\dots ,{\delta }_k^{i_s}],$$

$$W_{[m,n]}={\delta }_{mn}[1,\dots ,(n-1)m+1,\dots ,m,\dots ,nm],$$

and

$$J^{\eta }=\left[\begin{array}{c}{J}_1^{\eta }\\ ⋮\\ J_m^{\eta }\\ ⋮\\ J_n^{\eta }\end{array}\right],J_m^{\eta }=\left[\begin{array}{c}{J}_{1m}^{\eta }\\ ⋮\\ J_{rm}^{\eta }\\ ⋮\\ J_{nm}^{\eta }\end{array}\right],R^{\eta }=\left[\begin{array}{c}{R}_1^{\eta }\\ ⋮\\ R_m^{\eta }\\ ⋮\\ R_n^{\eta }\end{array}\right],R_m^{\eta }=\left[\begin{array}{c}{R}_{1m}^{\eta }\\ ⋮\\ R_{rm}^{\eta }\\ ⋮\\ R_{nm}^{\eta }\end{array}\right],m=1,2,\dots ,n,$$

where

$$\begin{array}{cc}\hfill & \mathrm{for}\eta =\mathbf{i},J_{rm}^{\mathbf{i}}=\left\{\begin{array}{cc}{\left({\delta }_{n(n+1)/2}^{{\displaystyle \frac{(r-1)(2n-r+2)}{2}}+m-r+1}\right)}^T\otimes R_4,\hfill & r<m,\hfill \\ {\left({\delta }_{n(n+1)/2}^{{\displaystyle \frac{(m-1)(2n-m+2)}{2}}+r-m+1}\right)}^T\otimes I_4,\hfill & r\ge m,\hfill \end{array}\right.R_4=\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& -1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right];\hfill \\ \hfill & \mathrm{for}\eta =\mathbf{j},J_{rm}^{\mathbf{j}}=\left\{\begin{array}{cc}{\left({\delta }_{n(n+1)/2}^{{\displaystyle \frac{(r-1)(2n-r+2)}{2}}+m-r+1}\right)}^T\otimes L_4,\hfill & r<m,\hfill \\ {\left({\delta }_{n(n+1)/2}^{{\displaystyle \frac{(m-1)(2n-m+2)}{2}}+r-m+1}\right)}^T\otimes I_4,\hfill & r\ge m,\hfill \end{array}\right.L_4=\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& -1& 0\\ 0& 0& 0& 1\end{array}\right];\hfill \\ \hfill & \mathrm{for}\eta =\mathbf{k},J_{rm}^{\mathbf{k}}=\left\{\begin{array}{cc}{\left({\delta }_{n(n+1)/2}^{{\displaystyle \frac{(r-1)(2n-r+2)}{2}}+m-r+1}\right)}^T\otimes S_4,\hfill & r<m,\hfill \\ {\left({\delta }_{n(n+1)/2}^{{\displaystyle \frac{(m-1)(2n-m+2)}{2}}+r-m+1}\right)}^T\otimes I_4,\hfill & r\ge m,\hfill \end{array}\right.S_4=\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& -1\end{array}\right];\hfill \end{array}$$

$$\begin{array}{cc}\hfill & \mathrm{for}\eta =\mathbf{i},R_{rm}^{\mathbf{i}}=\left\{\begin{array}{cc}{\left({\delta }_{n(n+1)/2}^{{\displaystyle \frac{(r-1)(2n-r+2)}{2}}+m-r+1}\right)}^T\otimes R_4^{\prime },\hfill & r<m,\hfill \\ {\left({\delta }_{n(n+1)/2}^{{\displaystyle \frac{(m-1)(2n-m+2)}{2}}+r-m+1}\right)}^T\otimes I_4,\hfill & r\ge m,\hfill \end{array}\right.R_4^{\prime }=\left[\begin{array}{cccc}-1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& -1& 0\\ 0& 0& 0& -1\end{array}\right];\hfill \\ \hfill & \mathrm{for}\eta =\mathbf{j},R_{rm}^{\mathbf{j}}=\left\{\begin{array}{cc}{\left({\delta }_{n(n+1)/2}^{{\displaystyle \frac{(r-1)(2n-r+2)}{2}}+m-r+1}\right)}^T\otimes L_4^{\prime },\hfill & r<m,\hfill \\ {\left({\delta }_{n(n+1)/2}^{{\displaystyle \frac{(m-1)(2n-m+2)}{2}}+r-m+1}\right)}^T\otimes I_4,\hfill & r\ge m,\hfill \end{array}\right.L_4^{\prime }=\left[\begin{array}{cccc}-1& 0& 0& 0\\ 0& -1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& -1\end{array}\right];\hfill \\ \hfill & \mathrm{for}\eta =\mathbf{k},R_{rm}^{\mathbf{k}}=\left\{\begin{array}{cc}{\left({\delta }_{n(n+1)/2}^{{\displaystyle \frac{(r-1)(2n-r+2)}{2}}+m-r+1}\right)}^T\otimes S_4^{\prime },\hfill & r<m,\hfill \\ {\left({\delta }_{n(n+1)/2}^{{\displaystyle \frac{(m-1)(2n-m+2)}{2}}+r-m+1}\right)}^T\otimes I_4,\hfill & r\ge m,\hfill \end{array}\right.S_4^{\prime }=\left[\begin{array}{cccc}-1& 0& 0& 0\\ 0& -1& 0& 0\\ 0& 0& -1& 0\\ 0& 0& 0& 1\end{array}\right].\hfill \end{array}$$

Theorem 14 

(Theorem 4.3 and Corollary 4.4, [82]). Let A, B, C, D, and E be given in Equation (26), and let

$$\widehat{M}=\left[\begin{array}{cc}{M}_1& M_2\end{array}\right],$$

where

$$\begin{array}{cc}\hfill M_1& =G_2⋉G_3⋉v_r^R\left(A\right)⋉W_{[4ns,4n^2]}⋉v_c^R\left(B\right)⋉J^{\eta },\hfill \\ \hfill M_2& =G_4⋉G_5⋉v_r^R\left(C\right)⋉W_{[4ks,4k^2]}⋉v_c^R\left(D\right)⋉R^{\eta },\hfill \end{array}$$

and $G_i$ has the same structure as G given in Remark 19 but differs in orders.

(1) 

Then, Equation (26) is consistent if and only if

$$\left(\widehat{M}{\widehat{M}}^{†}-I_{4ms}\right)v_r^R\left(E\right)=0.$$

(2) 

Let

$$S_M=\left\{(X,Y)|X={X^{\eta }}^{*},Y=-{Y^{\eta }}^{*},\parallel AXB+CYD-E\parallel =min\right\}.$$

Then,

$$S_M=\left\{(X,Y)|\left[\begin{array}{c}{v}_s^R\left(X\right)\\ v_s^R\left(Y\right)\end{array}\right]={\widehat{M}}^{†}{v}_r^R\left(E\right)+\left(I_{2(n^2+k^2)+2(n+k)}-{\widehat{M}}^{†}\widehat{M}\right)y,y\in {\mathbb{R}}^{2(n^2+k^2)+2(n+k)}\right\}.$$

(3) 

If $(\widehat{X},\widehat{Y})\in S_M$ satisfies

$$\parallel \widehat{X}{\parallel }^2+{\parallel \widehat{Y}\parallel }^2=\underset{(X,Y)\in S_M}{min}\left({\parallel X\parallel }^2+{\parallel Y\parallel }^2\right),$$

then

$$\left[\begin{array}{c}{v}_s^R\left(\widehat{X}\right)\\ v_s^R\left(\widehat{Y}\right)\end{array}\right]={\widehat{M}}^{†}{v}_r^R\left(E\right).$$

Remark 20. 

Setting B and C in Equation (26) as identity matrices, Theorem 14 yields the corresponding results for Equation (5) over $\mathbb{H}$.

Remark 21. 

During the same period, Ding et al. [83] and Wang et al. [84] also developed the real vector representation of quaternion matrices and applied this method to address problems in quaternion matrix equations. Inspired by this idea, Liu et al. [85] proposed the left and right real element representations of octonion matrices to solve the classical octonion matrix equation

$$AXB=C,$$

where X is unknown. Recently, Chen and Song [86] used the real vector representation of quaternion matrices to study the least-squares lower (upper) triangular Toeplitz solutions and (anti)centrosymmetric solutions of quaternion matrix equations.

Remark 22. 

It is known that real representations of quaternion matrices are not unique. For instance, Liu et al. [87] defined three real representations of quaternion matrices. Interestingly, Fan et al. first [88,89] proposed the $\mathcal{L}$-representation for quaternion matrices by STP to systematically study the real representations of quaternion matrices. Moreover, $\mathcal{L}$-representation serves as an effective tool in solving quaternion matrix equations. Meanwhile, Zhang et al. [90] also defined the $\mathcal{L}$-representation for commutative quaternion matrices and applied it to solving the corresponding matrix equations.

Following this idea, Fan et al. [91] established the $\mathcal{C}$-representation of quaternion matrices, which generalizes the complex representation of quaternion matrices. This new representation is also applied to study η-Hermitian solutions of the quaternion matrix equation

$$AXB=C,$$

with unknown X. Similarly, Xi et al. [92] defined the ${\mathcal{L}}_C$-representation of reduced biquaternion matrices to investigate the mixed solutions of the reduced biquaternion matrix equation

$$\sum _{i=1}^n{A}_i{X}_i{B}_i=E,$$

(27)

where $X_i(i=1,\dots ,n)$ is unknown. Evidently, Equation (5) is the special case of Equation (27) over reduced biquaternions.

Contemporaneously with [88,89], Fan et al. [93] also derived the minimal norm least-squares (anti)-Hermitian solution of the quaternion matrix equation directly by the vectorization properties of STP (i.e., Theorems 2.9 and 2.10, [93]) and the complex representation of quaternion matrix. Recently, Liu et al. [94] directly used the vectorization properties of STP to investigate (skew) bisymmetric solutions of the generalized Lyapunov quaternion matrix equation.

We contend that the four aforementioned STP-based methods—specifically, $\mathcal{L}$-representation, $\mathcal{C}$-representation, ${\mathcal{L}}_{\mathcal{C}}$-representation, and vectorization properties of STP—provide distinct perspectives for the investigation of Equation (1).

5. Constrained Solutions of GSE

When GSE fails to satisfy solvability conditions, find its least-squares solution under a certain matrix norm; since such solutions are always non-unique, further seeking for the minimum-norm least-squares solution (also known as the best approximate solution) is a common research approach. Furthermore, in practical applications, specific constraints are often imposed on the solutions of GSE (e.g., symmetric solutions, re-(non)positive definite solutions, equality-constrained solutions). Thus, this section focuses on various solutions to GSE.

5.1. Chebyshev Solutions and $l_p$-Solutions

Let $\mathbb{F}=\mathbb{R}$. Ziętak studied the Chebyshev solutions [95] and the $l_p$-solutions [96] of Equation (5) by using the Chebyshev norm and the $l_p$-norm of a matrix, respectively.

Definition 5. 

Let $A=\left[a_{i,j}\right]\in {\mathbb{R}}^{m\times n}$ and $1<p<\infty$. The Chebyshev norm of A, denoted by ${\parallel A\parallel }_{\infty }$, is defined as

$${\parallel A\parallel }_{\infty }=\underset{1\le i\le m,1\le j\le n}{max}\left|a_{i,j}\right|,$$

where $|a_{i,j}|$ is the absolute value of $a_{i,j}$ for $1\le i\le m$ and $1\le j\le n$. And, the $l_p$-norm of A, denoted by ${\parallel A\parallel }_p$, is defined as

$${\parallel A\parallel }_p={\left(\sum _{i=1}^m\sum _{j=1}^n{\left|a_{i,j}\right|}^p\right)}^{1/p}.$$

Theorem 15 

(Theorem 2.2, [95]). Let $r<m$ and $s<n$. Suppose that (4) is not satisfied. Then, the matrices $X_{\infty }=\left[x_{kj}\right]$ and $Y_{\infty }=\left[y_{il}\right]$ are a Chebyshev solution pair of Equation (5), i.e.,

$$\parallel A{X}_{\infty }+Y_{\infty }{B-C\parallel }_{\infty }=\underset{X,Y}{min}{\parallel AX+YB-C\parallel }_{\infty },$$

if and only if there exists $V=\left[v_{ij}\right](\ne 0)\in {\mathbb{R}}^{m\times n}$ such that $V^{T}A=0$, $V{B}^T=0$,

$$\begin{array}{cc}\hfill & v_{ij}\mathrm{sign}\left(r_{ij}\right)>0\mathit{for}(i,j)\in {\mathcal{J}}_1,\mathit{and}{v}_{ij}=0\mathit{for}(i,j)\notin {\mathcal{J}}_1,\hfill \end{array}$$

where

$$r_{ij}=\sum _{k=1}^r{a}_{ik}{x}_{kj}+\sum _{l=1}^s{y}_{il}{b}_{lj}-c_{ij},$$

and ${\mathcal{J}}_1$ is an appropriate subset of $\mathcal{J}=\left\{(i,j)\mid |r_{ij}{|=\parallel AX+YB-C\parallel }_{\infty }\right\}$.

Remark 23. 

Moreover, Ziętak formulated the equivalent conditions for the Chebyshev solution of Equation (5) by Theorem 3.3 of [95] and Theorem 4.1 of [95] under the assumption:

$$m=r+1,n=s+1,\mathrm{rank}\left(A\right)=r,\mathrm{rank}\left(B\right)=s,$$

(28)

and another assumption:

$$A\ne 0,B\ne 0,r=s=1,$$

(29)

respectively.

Theorem 16 

(Theorem 2.1, [96]). Let $r<m$, $s<n$, and $1<p<\infty$. Then, the matrices $X_p=\left[x_{i,j}\right]$ and $Y_p=\left[y_{i,j}\right]$ are an $l_p$-solution of Equation (5), i.e.,

$$\parallel A{X}_p+Y_p{B-C\parallel }_p=\underset{X,Y}{min}{\parallel AX+YB-C\parallel }_p,$$

if and only if

$$V^{T}A=0\mathit{and}V{B}^T=0,$$

where

$$\begin{array}{c}\hfill V={\left[v_{i,j}\right]}_{m\times n}={\left[\mathrm{sign}\left(r_{i,j}\right){\left|r_{i,j}\right|}^{p-1}\right]}_{m\times n}\mathit{and}{r}_{i,j}=\sum _{k=1}^r{a}_{i,k}{x}_{k,j}+\sum _{l=1}^s{y}_{i,l}{b}_{l,j}-c_{i,j}\end{array}$$

for $i=1,2,\dots ,m$ and $j=1,2,\dots ,n$.

Ziętak [96] further presented additional characterizations of the $l_p$-solutions of Equation (5). Based on these characterizations, an algorithm was designed for solving such solutions (see Section 7 for details).

Theorem 17 

(Theorems 2.2 and 2.3, [96]). Let A, B, and C be given in Equation (5) and let $1<p<\infty$. Then, the following are equivalent:

(1) 

The matrices $X_{*}\in {\mathbb{R}}^{r\times n}$ and $Y_{*}\in {\mathbb{R}}^{m\times s}$ are the $l_p$-solution of Equation (5);

(2) 

The following equalities hold:

$$\underset{X}{min}\parallel AX+Y_{*}{B-C\parallel }_p=\underset{Y}{min}\parallel A{X}_{*}{+YB-C\parallel }_p={\parallel A{X}_{*}+Y_{*}B-C\parallel }_p;$$

(3) 

The columns $x_j^{(*)}$ of $X_{*}\in {\mathbb{R}}^{r\times n}$ are the $l_p$-solutions of the linear systems

$$Ax=c_j-Y_{*}{b}_j(j=1,2,\dots ,n)$$

and the columns $y_i^{(*)}$ of $Y_{*}^T\in {\mathbb{R}}^{s\times m}$ are the $l_p$-solutions of the linear systems

$$B^{T}y=d_i-X_{*}^T{a}_i(i=1,2,\dots ,m),$$

where $x\in {\mathbb{R}}^r$, $y\in {\mathbb{R}}^s$, $c_j$, $b_j$, $d_i$ and $a_i$ are the appropriate columns of C, B, $C^T$, and $A^T$, respectively.

Remark 24. 

Since

$$A(X+(I-A^{-}A)W-A^{-}ZB)+(Y+Z(I-B{B}^{-})+A{A}^{-}ZB{B}^{-})B=AX+YB$$

for arbitrary W and Z, neither the Chebyshev solution nor the $l_p$-solution of Equation (5) is unique.

Remark 25. 

Note that, in Theorem 2.1 of [27], Xu et al. gave the explicit expression of the $l_2$-solutions of Equation (12). Moreover, in Theorems 4.2 and 4.3 of [97], Liao et al. also consider the best approximate solution of (12) to a given matrix pair $(X_f,Y_f)$ by using GSVD and CCD. Therefore, when both E and F are identity matrices, we can immediately obtain decomposed expressions of $l_2$-solutions and the best approximate solution of Equation (5).

5.2. ★-Congruent Solutions

We now discuss Equation (5) under the constraint condition $Y=X^{★}$, i.e.,

$$AX+X^{★}B=C,$$

(30)

where $X^{★}$ denotes either $X^T$ or $X^{*}$. We call X satisfying Equation (30) a ★-congruent solution of Equation (5).

Wimmer is the first to study the necessary and sufficient conditions for the solvability of Equation (30) under $★=*$ over $\mathbb{C}$ (see Theorem 2, [98]). After that, De Terán and Dopico [99] generalize the Wimmer’s work to a field $\mathbb{F}$ with $\mathrm{char}\left(\mathbb{F}\right)\ne 2$, in which case, $X^{★}$ denotes the transpose of X, except in the particular case $\mathbb{F}=\mathbb{C}$, where it may be either the transpose or the conjugate transpose of X. Moreover, we call $A\in {\mathbb{F}}^{n\times n}$ and $B\in {\mathbb{F}}^{n\times n}$ are ★-congruent if there exists a nonsingular matrix $P\in {\mathbb{F}}^{n\times n}$ such that $P^{★}AP=B$.

Theorem 18 

(Theorem 2.3, [99]). Let $\mathbb{F}$ be a field with $\mathrm{char}\left(\mathbb{F}\right)\ne 2$. Then, Equation (30) is solvable if and only if

$$\left[\begin{array}{cc}C& A\\ B& 0\end{array}\right]\mathit{and}\left[\begin{array}{cc}0& A\\ B& 0\end{array}\right]$$

are ★-congruent.

Remark 26. 

Additionally, in Lemma 5.10 of [100], Byers and Kressner established the equivalent conditions for the existence of a unique solution to Equation (30) only when $★=T$. Kressner et al. generalized this result in Lemma 8 of [101] to include the case where $★=*$. In Theorems 2.1 and 2.2 of [102], Cvetković-Ilić investigated Equation (30) for the bounded linear operators under certain conditions, in which case $X^{★}$ denotes the adjoint operator of X.

5.3. (Minimum-Norm Least-Squares) Symmetric Solutions

Let $\mathbb{F}=\mathbb{R}$, $s=m$, $r=n$, and $A=B$ for Equation (5), i.e.,

$$AX+YA=C.$$

(31)

Chang and Wang [103] studied the symmetric, minimum-2-norm symmetric, least-squares symmetric, and minimum-2-norm least-squares symmetric solutions of Equation (31) by SVD over $\mathbb{R}$. Let ${\mathbb{SR}}^{m\times m}$ be the set of all $m\times m$ real symmetric matrices, and let

$$A\circ B=\left[a_{ij}{b}_{ij}\right]\in {\mathbb{F}}^{m\times n}$$

denote the Hadamard product of $A=\left[a_{ij}\right]\in {\mathbb{F}}^{m\times n}$ and $B=\left[b_{ij}\right]\in {\mathbb{F}}^{m\times n}$.

Theorem 19 

(Theorem 2.1, [103]). Let the SVD of A be

$$A=U\left[\begin{array}{cc}\mathsf{\Sigma }& 0\\ 0& 0\end{array}\right]V^T,$$

where $\mathsf{\Sigma }=\mathrm{diag}({\sigma }_1,\dots ,{\sigma }_r)>0$, $r=\mathrm{rank}\left(A\right)$, and

$$U=\left[\begin{array}{cc}{U}_1& U_2\end{array}\right]\in {\mathbb{R}}^{m\times m}\mathit{and}V=\left[\begin{array}{cc}{V}_1& V_2\end{array}\right]\in {\mathbb{R}}^{n\times n}$$

are real orthogonal with

$$U_1\in {\mathbb{R}}^{m\times r},U_2\in {\mathbb{R}}^{m\times (m-r)},V_1\in {\mathbb{R}}^{n\times r},\mathit{and}{V}_2\in {\mathbb{R}}^{n\times (n-r)}.$$

Denote

$$\begin{array}{cc}\hfill & W_1=\mathsf{\Sigma }{U}_1^{T}C{V}_1-V_1^T{C}^T{U}_1\mathsf{\Sigma },W_2={\mathsf{\Sigma }}^{-1}\left(U_1^{T}C{V}_1+V_1^T{C}^T{U}_1\right),\hfill \\ \hfill & M_1=\mathsf{\Sigma }{V}_1^T{C}^T{U}_1-U_1^{T}C{V}_1\mathsf{\Sigma },M_2={\mathsf{\Sigma }}^{-1}{U}_1^{T}C{V}_1.\hfill \end{array}$$

Define

$$\Phi =\left[{\phi }_{ij}\right]\in {\mathbb{R}}^{r\times r}\mathit{and}\mathsf{\Psi }=\left[{\psi }_{ij}\right]\in {\mathbb{R}}^{r\times r},$$

where

$${\phi }_{ij}=\left\{\begin{array}{cc}{\displaystyle \frac{1}{{\sigma }_i^2-{\sigma }_j^2}},\hfill & {\sigma }_i\ne {\sigma }_j,\hfill \\ 0,\hfill & {\sigma }_i={\sigma }_j,\hfill \end{array}\right.\mathrm{and}{\psi }_{ij}=1-\left|\mathrm{sign}({\sigma }_i-{\sigma }_j)\right|$$

for $i,j=1,\dots ,r$.

(1) 

Let

$${\mathcal{L}}_I=\{[X,Y]\mid X\in {\mathbb{SR}}^{n\times n},Y\in {\mathbb{SR}}^{m\times m},AX+YA=C\}.$$

Then, ${\mathcal{L}}_I\ne \varnothing$ if and only if

$$U_2^{T}C{V}_2=0\mathit{and}\mathsf{\Psi }\circ \left(U_1^{T}C{V}_1-V_1^T{C}^T{U}_1\right)=0,$$

in which case

$$\begin{array}{cc}\hfill {\mathcal{L}}_I=\left\{\right[V& \left[\begin{array}{cc}\Phi \circ W_1+\mathsf{\Psi }\circ \left(M_2-Y_{11}\right)& {\mathsf{\Sigma }}^{-1}{U}_1^{T}C{V}_2\\ V_2^T{C}^T{U}_1{\mathsf{\Sigma }}^{-1}& X_{22}\end{array}\right]V^T,\hfill \\ \hfill U& \left[\begin{array}{cc}\Phi \circ M_1+\mathsf{\Psi }\circ Y_{11}& {\mathsf{\Sigma }}^{-1}{V}_1^T{C}^T{U}_2\\ U_2^{T}C{V}_1{\mathsf{\Sigma }}^{-1}& Y_{22}\end{array}\right]U^T]\hfill \\ \hfill & \mid X_{22}\in {\mathbb{SR}}^{(n-r)\times (n-r)},Y_{11}\in {\mathbb{SR}}^{r\times r},Y_{22}\in {\mathbb{SR}}^{(m-r)\times (m-r)}\}.\hfill \end{array}$$

(2) 

If $[\widehat{X},\widehat{Y}]\in {\mathcal{L}}_I$ satisfies

$$\parallel [\widehat{X},\widehat{Y}]{\parallel }_F=\underset{[X,Y]\in {\mathcal{L}}_I}{min}(\parallel \widehat{X}{\parallel }_F^2+\parallel \widehat{Y}{{\parallel }_F^2)}^{{\displaystyle \frac{1}{2}}},$$

then $[\widehat{X},\widehat{Y}]\in {\mathcal{L}}_I$ is unique and

$$\begin{array}{cc}\hfill \widehat{X}& =V\left[\begin{array}{cc}\Phi \circ W_1+{\displaystyle \frac{1}{2}}\mathsf{\Psi }\circ M_2& {\mathsf{\Sigma }}^{-1}{U}_1^{T}C{V}_2\\ V_2^T{C}^T{U}_1{\mathsf{\Sigma }}^{-1}& 0\end{array}\right]V^T,\hfill \\ \hfill \widehat{Y}& =U\left[\begin{array}{cc}\Phi \circ M_1+{\displaystyle \frac{1}{2}}\mathsf{\Psi }\circ M_2& {\mathsf{\Sigma }}^{-1}{V}_1^T{C}^T{U}_2\\ U_2^{T}CV{\mathsf{\Sigma }}^{-1}& 0\end{array}\right]U^T.\hfill \end{array}$$

(3) 

Let

$${\mathcal{L}}_{ILS}=\left\{[X,Y]\right|X\in {\mathbb{SR}}^{n\times n},Y\in {\mathbb{SR}}^{m\times m}{,\parallel AX+YA-C\parallel }_F=min\}.$$

Then,

$$\begin{array}{cc}\hfill {\mathcal{L}}_{ILS}=\left\{\right[V& \left[\begin{array}{cc}\Phi \circ W_1+\mathsf{\Psi }\circ ({\displaystyle \frac{1}{2}}{W}_2-Y_{11})& {\mathsf{\Sigma }}^{-1}{U}_1^{T}C{V}_2\\ V_2^T{C}^T{U}_1{\mathsf{\Sigma }}^{-1}& X_{22}\end{array}\right]V^T,\hfill \\ \hfill U& \left[\begin{array}{cc}\Phi \circ M_1+\mathsf{\Psi }\circ Y_{11}& {\mathsf{\Sigma }}^{-1}{V}_1^T{C}^T{U}_2\\ U_2^{T}C{V}_1{\mathsf{\Sigma }}^{-1}& Y_{22}\end{array}\right]U^T]\hfill \\ \hfill & \mid X_{22}\in {\mathbb{SR}}^{(n-r)\times (n-r)},Y_{11}\in {\mathbb{SR}}^{r\times r},Y_{22}\in {\mathbb{SR}}^{(m-r)\times (m-r)}\}.\hfill \end{array}$$

(4) 

If $[\widehat{X},\widehat{Y}]\in {\mathcal{L}}_{ILS}$ satisfies

$$\parallel [\widehat{X},\widehat{Y}]{\parallel }_F=\underset{[X,Y]\in {\mathcal{L}}_{ILS}}{min}(\parallel \widehat{X}{\parallel }_F^2+\parallel \widehat{Y}{{\parallel }_F^2)}^{{\displaystyle \frac{1}{2}}},$$

then $[\widehat{X},\widehat{Y}]$ is unique and

$$\begin{array}{cc}\hfill \widehat{X}& =V\left[\begin{array}{cc}\Phi \circ W_1+{\displaystyle \frac{1}{4}}\mathsf{\Psi }\circ W_2& {\mathsf{\Sigma }}^{-1}{U}_1^{T}C{V}_2\\ V_2^T{C}^T{U}_1{\mathsf{\Sigma }}^{-1}& 0\end{array}\right]V^T,\hfill \\ \hfill \widehat{Y}& =U\left[\begin{array}{cc}\Phi \circ M_1+{\displaystyle \frac{1}{4}}\mathsf{\Psi }\circ W_2& {\mathsf{\Sigma }}^{-1}{V}_1^T{C}^T{U}_2\\ U_2^{T}C{V}_1{\mathsf{\Sigma }}^{-1}& 0\end{array}\right]U^T.\hfill \end{array}$$

5.4. Self-Adjoint and Positive (Semi)Definite Solutions

In this subsection, we consider Equation (1) with $r=n$, $s=m$ and $C=0$, i.e.,

$$AX-YB=0,$$

(32)

which is required in optimal control theory [104].

Jameson et al. [105] explored the symmetric, positive semidefinite, and positive definite real solutions of Equation (32) over $\mathbb{R}$. Subsequently, Dobovišek [106] further studied the self-adjoint, positive semidefinite, positive definite, minimal, and extreme solutions of Equation (32) over $\mathbb{C}$.

Definition 6 

([106]). If the nonnegative matrix $Y_m$ is such that $(X,Y_m)$ is a solution pair of Equation (32), and $Y_m\le Y$ (or $Y_m\ge Y$) for all other solution pairs $(X,Y)$ of Equation (32) with the same X, then $Y_m$ is called a minimal (or maximal) solution of Equation (32). The minimal and maximal solutions of Equation (32) are collectively referred to as the extreme solutions of Equation (32).

Theorem 20 

([106]). Let A be of full column rank, $m\ge n$,

$$A=\left[\begin{array}{c}0\\ I\end{array}\right],B=\left[\begin{array}{c}{B}_1\\ B_2\end{array}\right],\mathit{and}Y=\left[\begin{array}{cc}{Y}_{11}& Y_{12}\\ Y_{12}^{*}& Y_{22}\end{array}\right].$$

(1) 

(Theorem 6, [106]) Assume that $A{B}^{*}$ has at least one real eigenvalue or at least one conjugate pair of eigenvalues. Then, Equation (32) has a non-zero self-adjoined solution pair, i.e., $X^{*}=X$ and $Y^{*}=Y$.

(2) 

(Theorem 7, [106]) Equation (32) has a nonzero positive semidefinite solution Y, i.e., $Y\ge 0$, if and only if $A{B}^{*}$ has at least one real eigenvalue.

(3) 

(Theorem 8, [106]) Equation (32) has a positive definite solution Y, i.e., $Y>0$, if and only if $A{B}^{*}$ has only real eigenvalues and is diagonalizable.

(4) 

(Theorem 9, [106]) Equation (32) has a positive semidefinite solution, i.e., $X\ge 0$, if and only if the matrix A has at least one real eigenvalue, in which case, $X\ge 0$ is nonzero if $\mathrm{rank}\left(A{B}^{*}\right)>\mathrm{rank}\left(B_2\right)$.

(5) 

(Theorem 10, [106]) Equation (32) has a positive definite solution, i.e., $X>0$, if and only if all eigenvalues of $B_2^{*}$ are real, $B_2^{*}$ is diagonalizable, and $\mathrm{rank}\left(A{B}^{*}\right)=\mathrm{rank}\left(A\right)$. Moreover, if $Y\ge 0$, then $X>0$ if and only if all eigenvalues of $B^{*}$ are positive and $B^{*}$ is diagonalizable.

(6) 

(Theorem 11, [106]) If $\mathrm{rank}\left(B\right)<m$, then Equation (32) with a fixed solution X does not have an extreme solution for Y. If $\mathrm{rank}\left(B\right)=m$, the solution Y is unique.

(7) 

(Theorem 12, [106]) If Equation (32) has a solution pair $(X,Y)$ and $Y\ge 0$, then there exists a minimal solution $Y_m\ge 0$.

Remark 27. 

The expressions of self-adjoint, positive semidefinite, and positive definite solutions are also presented when the solvability conditions are met (see (Theorems 6–10, [106])). Additionally, Dobovišek discussed these solutions for $m<n$ (see (Theorems 13–17, [106])) and for matrices A and B without full rank (see (Theorems 18–20, [106])).

5.5. Per(Skew)Symmetric and Bi(Skew)Symmetric Solutions

It is well known that (skew)selfconjugate, per(skew)symmetric, and centro(skew) symmetric matrices be applied to information theory, linear system theory, and numerical analysis theory (see [107,108,109,110]). Let $\mathbb{F}=\Omega$ be a finite dimensional central algebra with an involution $\sigma$ and $\mathrm{char}(\Omega )\ne 2$ (see [111]). Wang et al. in [112,113] gave necessary and sufficient conditions for the existence of per(skew)symmetric solutions and bi(skew)symmetric solutions to Equation (1) over $\Omega$, respectively.

Definition 7. 

For $A=\left[a_{i,j}\right]\in {\Omega }^{m\times n}$, let

$$A^{*}=\left[\sigma \left(a_{i,j}\right)\right]\in {\Omega }^{n\times m},A^{(*)}=\left[\sigma \left(a_{n-j+1,m-i+1}\right)\right]\in {\Omega }^{n\times m},A^{\#}=\left[a_{m-i+1,n-j+1}\right]\in {\Omega }^{m\times n}.$$

Then, A is called to be (skew)selfconjugate if $A=A^{*}(-A^{*})$, per(skew)symmetric if $A=A^{(*)}(-A^{(*)})$, and centro(skew)symmetric if $A=A^{\#}(-A^{\#})$. If A is both (skew)selfconjugate and per(skew)symmetric, then A is called to be bi(skew)symmetric. Moreover, a solution pair $(X,Y)$ of Equation (1) is called to be per(skew)symmetric (or bi(skew)symmetric) if both X and Y are per(skew)symmetric (or bi(skew)symmetric).

Theorem 21 

([112]). Let $A,B,C\in {\Omega }^{m\times n}\left[\lambda \right]$.

(1) 

(Corollary 2.3, [112]) Equation (1) has a persymmetric solution pair $(X,Y)$ if and only if there exist invertible matrices $P\in {\Omega }^{2n\times 2n}$ and $Q\in {\Omega }^{2m\times 2m}$ such that

$$\begin{array}{cc}\hfill Q\left[\begin{array}{cc}A& C\\ 0& B\end{array}\right]P^{-1}& =\left[\begin{array}{cc}A& 0\\ 0& B\end{array}\right],\hfill \end{array}$$

(33)

$$\begin{array}{cc}\hfill P\left[\begin{array}{cc}-I_n& 0\\ 0& I_n\end{array}\right]P^{(*)}& =\left[\begin{array}{cc}-I_n& 0\\ 0& I_n\end{array}\right],\hfill \end{array}$$

(34)

$$\begin{array}{cc}\hfill Q\left[\begin{array}{cc}-I_m& 0\\ 0& I_m\end{array}\right]Q^{(*)}& =\left[\begin{array}{cc}-I_m& 0\\ 0& I_m\end{array}\right].\hfill \end{array}$$

(35)

(2) 

(Corollary 2.8, [112]) Equation (1) has a perskewsymmetric solution pair $(X,Y)$ if and only if there exist invertible matrices $P\in {\Omega }^{2n\times 2n}$ and $Q\in {\Omega }^{2m\times 2m}$ such that (33), $P^{(*)}P=I_{2n}$, and $Q^{(*)}Q=I_{2m}$.

(3) 

(Corollary 2.13, [112]) Equation (1) has a solution pair $(X,Y)$ such that X is persymmetric and Y is perskewsymmetric if and only if there exist invertible matrices $P\in {\Omega }^{2n\times 2n}$ and $Q\in {\Omega }^{2m\times 2m}$ such that (33), (34), and $Q^{(*)}Q=I_{2m}$.

(4) 

(Corollary 2.16, [112]) Equation (1) has a solution pair $(X,Y)$ such that X is perskewsymmetric and Y is persymmetric if and only if there exist invertible matrices $P\in {\Omega }^{2n\times 2n}$ and $Q\in {\Omega }^{2m\times 2m}$ such that (33), (35), and $P^{(*)}P=I_{2n}$.

Theorem 22 

([113]). Let $A,B,C\in {\Omega }^{m\times n}\left[\lambda \right]$.

(1) 

(Corollary 2, [113]) Equation (1) has a bisymmetric solution pair $(X,Y)$ if and only if there exist invertible matrices $P\in {\Omega }^{2n\times 2n}$ and $Q\in {\Omega }^{2m\times 2m}$ such that

$$\begin{array}{cc}\hfill Q\left[\begin{array}{cc}A& C\\ 0& B\end{array}\right]P^{-1}& =\left[\begin{array}{cc}A& 0\\ 0& B\end{array}\right],\hfill \end{array}$$

(36)

$$\begin{array}{cc}\hfill P\left[\begin{array}{cc}0& I_n\\ -I_n& 0\end{array}\right]P^{*}& =\left[\begin{array}{cc}0& I_n\\ -I_n& 0\end{array}\right],\hfill \end{array}$$

(37)

$$\begin{array}{cc}\hfill P\left[\begin{array}{cc}-I_n& 0\\ 0& I_n\end{array}\right]P^{(*)}& =\left[\begin{array}{cc}-I_n& 0\\ 0& I_n\end{array}\right],\hfill \end{array}$$

(38)

$$\begin{array}{cc}\hfill P^{\#}\left[\begin{array}{cc}0& I_n\\ I_n& 0\end{array}\right]P^{-1}& =\left[\begin{array}{cc}0& I_n\\ I_n& 0\end{array}\right],\hfill \end{array}$$

(39)

$$\begin{array}{cc}\hfill Q\left[\begin{array}{cc}0& I_m\\ -I_m& 0\end{array}\right]Q^{*}& =\left[\begin{array}{cc}0& I_m\\ -I_m& 0\end{array}\right],\hfill \end{array}$$

(40)

$$\begin{array}{cc}\hfill Q\left[\begin{array}{cc}-I_m& 0\\ 0& I_m\end{array}\right]Q^{(*)}& =\left[\begin{array}{cc}-I_m& 0\\ 0& I_m\end{array}\right],\hfill \end{array}$$

(41)

$$\begin{array}{cc}\hfill Q^{\#}\left[\begin{array}{cc}0& I_m\\ I_m& 0\end{array}\right]Q^{-1}& =\left[\begin{array}{cc}0& I_m\\ I_m& 0\end{array}\right].\hfill \end{array}$$

(42)

(2) 

(Corollary 5, [113]) Equation (1) has a biskewsymmetric solution pair $(X,Y)$ if and only if there exist invertible matrices $P\in {\Omega }^{2n\times 2n}$ and $Q\in {\Omega }^{2m\times 2m}$ such that (36),

$$\begin{array}{cc}\hfill & P^{(*)}P=I_{2n},P\left[\begin{array}{cc}0& I_n\\ I_n& 0\end{array}\right]P^{*}=\left[\begin{array}{cc}0& I_n\\ I_n& 0\end{array}\right]=P^{\#}\left[\begin{array}{cc}0& I_n\\ I_n& 0\end{array}\right]P^{-1},\hfill \end{array}$$

(43)

$$\begin{array}{cc}\hfill & Q^{(*)}Q=I_{2m},Q\left[\begin{array}{cc}0& I_m\\ I_m& 0\end{array}\right]Q^{*}=\left[\begin{array}{cc}{I}_m& 0\\ 0& I_m\end{array}\right]=Q^{\#}\left[\begin{array}{cc}0& I_m\\ I_m& 0\end{array}\right]Q^{-1}.\hfill \end{array}$$

(44)

(3) 

(Corollary 8, [113]) Equation (1) has a solution pair $(X,Y)$ such that X is bisymmetric and Y is biskewsymmetric if and only if there exist invertible matrices $P\in {\Omega }^{2n\times 2n}$ and $Q\in {\Omega }^{2m\times 2m}$ such that (36)–(39), and (44).

(4) 

(Corollary 10, [113]) Equation (1) has a solution pair $(X,Y)$ such that X is biskewsymmetric and Y is bisymmetric if and only if there exist invertible matrices $P\in {\Omega }^{2n\times 2n}$ and $Q\in {\Omega }^{2m\times 2m}$ such that (36), (40)–(42), and (43).

5.6. Maximal and Minimal Ranks of the General Solution

Let $\mathbb{F}=\mathbb{C}$, and let a solution pair $(X,Y)$ of Equation (5) be

$$X=X_0+X_1\mathbf{i}\in {\mathbb{C}}^{r\times n}\mathrm{and}Y=Y_0+Y_1\mathbf{i}\in {\mathbb{C}}^{m\times s},$$

where $X_0,X_1\in {\mathbb{R}}^{r\times n}$ and $Y_0,Y_1\in {\mathbb{R}}^{m\times s}$. Liu [40] determined the maximal and minimal ranks for X, Y, $X_0$, $X_1$, $Y_0$, and $Y_1$.

Theorem 23 

(Theorems 2.1 and 2.2, [40]). Let Equation (5) be consistent.

(1) 

Then,

$$\begin{array}{cc}\hfill \underset{AX+YB=C}{max}\mathrm{rank}\left(X\right)& =min\left\{n,r,r-\mathrm{rank}\left(A\right)+\mathrm{rank}\left[\begin{array}{c}B\hfill \\ C\hfill \end{array}\right]\right\},\hfill \\ \hfill \underset{AX+YB=C}{max}\mathrm{rank}\left(Y\right)& =min\{m,s,s-\mathrm{rank}(B)+\mathrm{rank}[A,C\left]\right\},\hfill \\ \hfill \underset{AX+YB=C}{min}\mathrm{rank}\left(X\right)& =\mathrm{rank}\left[\begin{array}{c}B\hfill \\ C\hfill \end{array}\right]-\mathrm{rank}\left(B\right),\hfill \\ \hfill \underset{AX+YB=C}{min}\mathrm{rank}\left(Y\right)& =\mathrm{rank}[A,C]-\mathrm{rank}\left(A\right).\hfill \end{array}$$

(2) 

Let

$$\begin{array}{cc}\hfill {\mathbb{S}}_1& =\left\{X_0\in {\mathbb{R}}^{r\times n}\mid A(X_0+i{X}_1)+(Y_0+i{Y}_1)B=C\right\},\hfill \\ \hfill {\mathbb{S}}_2& =\left\{X_1\in {\mathbb{R}}^{r\times n}\mid A(X_0+i{X}_1)+(Y_0+i{Y}_1)B=C\right\}.\hfill \end{array}$$

Then,

$$\begin{array}{cc}\hfill & \underset{X_0\in {\mathbb{S}}_1}{max}r\left(X_0\right)=min\left\{r,n,\mathrm{rank}\left[\begin{array}{cc}{B}_0& 0\\ B_1& 0\\ C_1& A_0\\ -C_0& A_1\end{array}\right]-2\mathrm{rank}\left(A\right)+r\right\},\hfill \\ \hfill & \underset{X_0\in {\mathbb{S}}_1}{min}r\left(X_0\right)=\mathrm{rank}\left[\begin{array}{cc}{B}_0& 0\\ B_1& 0\\ C_1& A_0\\ -C_0& A_1\end{array}\right]-\mathrm{rank}\left[\begin{array}{c}{A}_0\\ A_1\end{array}\right]-\mathrm{rank}\left[\begin{array}{c}{B}_0\hfill \\ B_1\hfill \end{array}\right],\hfill \end{array}$$

$$\begin{array}{cc}\hfill & \underset{X_1\in {\mathbb{S}}_2}{max}r\left(X_1\right)=min\left\{r,n,\mathrm{rank}\left[\begin{array}{cc}{B}_0& 0\\ B_1& 0\\ C_0& A_0\\ C_1& A_1\end{array}\right]-2\mathrm{rank}\left(A\right)+r\right\},\hfill \\ \hfill & \underset{X_1\in {\mathbb{S}}_2}{min}r\left(X_1\right)=\mathrm{rank}\left[\begin{array}{cc}{B}_0& 0\\ B_1& 0\\ C_0& A_0\\ C_1& A_1\end{array}\right]-\mathrm{rank}\left[\begin{array}{c}{A}_0\\ A_1\end{array}\right]-\mathrm{rank}\left[\begin{array}{c}{B}_0\\ B_1\end{array}\right].\hfill \end{array}$$

(3) 

Let

$$\begin{array}{cc}\hfill {\mathbb{S}}_3& =\left\{Y_0\in {\mathbb{R}}^{m\times s}\mid A(X_0+i{X}_1)+(Y_0+i{Y}_1)B=C\right\},\hfill \\ \hfill {\mathbb{S}}_4& =\left\{Y_1\in {\mathbb{R}}^{m\times s}\mid A(X_0+i{X}_1)+(Y_0+i{Y}_1)B=C\right\}.\hfill \end{array}$$

Then,

$$\begin{array}{cc}\hfill & \underset{Y_0\in {\mathbb{S}}_3}{max}r\left(Y_0\right)=min\left\{m,s,\mathrm{rank}\left[\begin{array}{cccc}{A}_0& A_1& C_1& -C_0\\ 0& 0& B_0& B_1\end{array}\right]-2\mathrm{rank}\left(B\right)+s\right\},\hfill \\ \hfill & \underset{Y_0\in {\mathbb{S}}_3}{min}r\left(Y_0\right)=\mathrm{rank}\left[\begin{array}{cccc}{A}_0& A_1& C_1& -C_0\\ 0& 0& B_0& B_1\end{array}\right]-\mathrm{rank}\left[A_0,A_1\right]-\mathrm{rank}\left[B_0,B_1\right],\hfill \\ \hfill & \underset{Y_1\in {\mathbb{S}}_4}{max}r\left(Y_1\right)=min\left\{m,s,\mathrm{rank}\left[\begin{array}{cccc}{A}_0& A_1& C_0& C_1\\ 0& 0& B_0& B_1\end{array}\right]-2\mathrm{rank}\left(B\right)+s\right\},\hfill \\ \hfill & \underset{Y_1\in {\mathbb{S}}_4}{min}r\left(Y_1\right)=\mathrm{rank}\left[\begin{array}{cccc}{A}_0& A_1& C_0& C_1\\ 0& 0& B_0& B_1\end{array}\right]-\mathrm{rank}\left[A_0,A_1\right]-\mathrm{rank}\left[B_0,B_1\right].\hfill \end{array}$$

Remark 28. 

In (Corollary 2.3, [40]), Liu also presented equivalent conditions for Equation (5) to have a (all) real solution pair(s), i.e.,

$$X=X_0\mathit{and}Y=Y_0,$$

and a (all) pure imaginary solution pair(s), i.e.,

$$X=\mathbf{i}{X}_1\mathit{and}Y=\mathbf{i}{Y}_1.$$

However, in (Section 3, [114]), Wang et al. provided two counterexamples to illustrate that the items (a) and (c) in (Corollary 2.3, [40]) are incorrect.

5.7. Re-(Non)negative and Re-(Non)positive Definite Solutions

Let $\mathbb{F}=\mathbb{C}$. For a Hermitian matrix $A\in {\mathbb{C}}^{n\times n}$, $i_{+}\left(A\right)$, $i_{-}\left(A\right)$, and $i_0\left(A\right)$ represent the numbers of the positive, negative, and zero eigenvalues of A, respectively. In Corollary 5.7 of [115], Wang and He established the maximal and minimal values of

$$i_{\pm }(X+X^{*})\mathrm{and}{i}_{\pm }(Y+Y^{*})$$

for a solution pair $(X,Y)$ of the complex matrix Equation

$$AXB+CYD=E,$$

(45)

where $A\in {\mathbb{C}}^{m\times k_2}$, $B\in {\mathbb{C}}^{k_2\times q}$, $C\in {\mathbb{C}}^{m\times k_3}$, $D\in {\mathbb{C}}^{k_3\times q}$, and $E\in {\mathbb{C}}^{m\times q}$ are given. This result directly yields the equivalent conditions for re-positive definite, re-negative definite, re-nonnegative definite, and re-nonpositive definite solutions to Equation (45).

Definition 8 

([115]). Let $A\in {\mathbb{C}}^{n\times n}$ and $H\left(A\right)=A+A^{*}$. The A is called to be re-positive definite if $H\left(A\right)>0$, re-nonnegative definite if $H\left(A\right)\ge 0$, re-negative definite if $H\left(A\right)<0$, and re-nonpositive definite if $H\left(A\right)\le 0$.

Theorem 24 

(Corollary 5.8, [115]). Let $X\in {\mathbb{C}}^{k_2\times k_2}$ and $Y\in {\mathbb{C}}^{k_3\times k_3}$ be a solution pair of Equation (45). Denote

$$\begin{array}{cc}\hfill G_1& =\left[\begin{array}{cccc}0& -D& C^{*}& 0\\ -D^{*}& 0& E^{*}& 0\\ C& E& 0& A\\ 0& 0& A^{*}& 0\end{array}\right],G_2=\left[\begin{array}{cccc}0& -C^{*}& D& 0\\ -C& 0& E& 0\\ D^{*}& E^{*}& 0& B^{*}\\ 0& 0& B& 0\end{array}\right],\hfill \\ \hfill G_3& =\left[\begin{array}{cccc}0& -B& A^{*}& 0\\ -B^{*}& 0& E^{*}& 0\\ A& E& 0& C\\ 0& 0& C^{*}& 0\end{array}\right],G_4=\left[\begin{array}{cccc}0& -A^{*}& B& 0\\ -A& 0& E& 0\\ B^{*}& E^{*}& 0& D^{*}\\ 0& 0& D& 0\end{array}\right],\hfill \end{array}$$

$$\begin{array}{cc}\hfill s_1& =\mathrm{rank}\left[\begin{array}{ccccc}0& -A^{*}& B& 0& 0\\ -A& 0& E& C& 0\\ B^{*}& E^{*}& 0& 0& D^{*}\end{array}\right],s_2=\mathrm{rank}\left[\begin{array}{ccccc}0& -B& A^{*}& 0& 0\\ -A& E& 0& C& 0\\ B^{*}& 0& E^{*}& 0& D^{*}\\ 0& 0& C^{*}& 0& 0\end{array}\right],\hfill \\ \hfill s_3& =\mathrm{rank}\left[\begin{array}{ccccc}0& -B& A^{*}& 0& 0\\ -A& E& 0& C& 0\\ B^{*}& 0& E^{*}& 0& D^{*}\\ 0& D& 0& 0& 0\end{array}\right],s_4=\mathrm{rank}\left[\begin{array}{c}A\\ B^{*}\end{array}\right],\hfill \end{array}$$

$$\begin{array}{cc}\hfill w_1& =\mathrm{rank}\left[\begin{array}{ccccc}0& -C^{*}& D& 0& 0\\ -C& 0& E& A& 0\\ D^{*}& E^{*}& 0& 0& B^{*}\end{array}\right],w_2=\mathrm{rank}\left[\begin{array}{ccccc}0& -D& C^{*}& 0& 0\\ -C& E& 0& A& 0\\ D^{*}& 0& E^{*}& 0& B^{*}\\ 0& 0& A^{*}& 0& 0\end{array}\right],\hfill \\ \hfill w_3& =\mathrm{rank}\left[\begin{array}{ccccc}0& -D& C^{*}& 0& 0\\ -C& E& 0& A& 0\\ D^{*}& 0& E^{*}& 0& B^{*}\\ 0& B& 0& 0& 0\end{array}\right],w_4=\mathrm{rank}\left[\begin{array}{c}C\\ D^{*}\end{array}\right].\hfill \end{array}$$

Then,

(1) 

X is re-positive definite if and only if

$$\begin{array}{cc}\hfill i_{+}\left(G_3\right)& =\mathrm{rank}\left[\begin{array}{cc}A\hfill & C\hfill \end{array}\right]+\mathrm{rank}\left(B\right)\mathit{and}{i}_{+}\left(G_3\right)\ge \mathrm{rank}\left[\begin{array}{c}D\hfill \\ B\hfill \end{array}\right]+\mathrm{rank}\left(A\right),\hfill \\ \hfill \mathit{or}{i}_{+}\left(G_3\right)& \ge \mathrm{rank}\left[\begin{array}{cc}A\hfill & C\hfill \end{array}\right]+\mathrm{rank}\left(B\right)\mathit{and}{i}_{+}\left(G_4\right)=\mathrm{rank}\left[\begin{array}{c}D\hfill \\ B\hfill \end{array}\right]+\mathrm{rank}\left(A\right).\hfill \end{array}$$

(2) 

X is re-negative definite if and only if

$$\begin{array}{cc}\hfill i_{-}\left(G_3\right)& =\mathrm{rank}\left[\begin{array}{cc}A\hfill & C\hfill \end{array}\right]+\mathrm{rank}\left(B\right)\mathit{and}{i}_{-}\left(G_4\right)\ge \mathrm{rank}\left[\begin{array}{c}D\hfill \\ B\hfill \end{array}\right]+\mathrm{rank}\left(A\right),\hfill \\ \hfill \mathit{or}{i}_{-}\left(G_3\right)& \ge \mathrm{rank}\left[\begin{array}{cc}A\hfill & C\hfill \end{array}\right]+\mathrm{rank}\left(B\right)\mathit{and}{i}_{-}\left(G_4\right)=\mathrm{rank}\left[\begin{array}{c}D\hfill \\ B\hfill \end{array}\right]+\mathrm{rank}\left(A\right).\hfill \end{array}$$

(3) 

X is re-nonnegative definite if and only if

$$\begin{array}{cc}\hfill s_1-s_4+i_{-}\left(G_3\right)-s_2& =0\mathit{and}{s}_1-s_4+i_{-}\left(G_4\right)-s_3\le 0,\hfill \\ \hfill \mathit{or}{s}_1-s_4+i_{-}\left(G_3\right)-s_2& \le 0\mathit{and}{s}_1-s_4+i_{-}\left(G_4\right)-s_3=0.\hfill \end{array}$$

(4) 

X is re-nonpositive definite if and only if

$$\begin{array}{cc}\hfill s_1-s_4+i_{+}\left(G_3\right)-s_2& =0\mathit{and}{s}_1-s_4+i_{+}\left(G_3\right)-s_3\le 0,\hfill \\ \hfill \mathit{or}{s}_1-s_4+i_{+}\left(G_3\right)-s_2& \le 0\mathit{and}{s}_1-s_4+i_{+}\left(G_4\right)-s_3=0.\hfill \end{array}$$

(5) 

Y is re-positive definite if and only if

$$\begin{array}{cc}\hfill i_{+}\left(G_1\right)& =\mathrm{rank}\left[\begin{array}{cc}A\hfill & C\hfill \end{array}\right]+\mathrm{rank}\left(D\right)\mathit{and}{i}_{+}\left(G_2\right)\ge \mathrm{rank}\left[\begin{array}{c}D\hfill \\ B\hfill \end{array}\right]+\mathrm{rank}\left(C\right),\hfill \\ \hfill \mathit{or}{i}_{+}\left(G_1\right)& \ge \mathrm{rank}\left[\begin{array}{cc}A\hfill & C\hfill \end{array}\right]+\mathrm{rank}\left(D\right)\mathit{and}{i}_{+}\left(G_2\right)=\mathrm{rank}\left[\begin{array}{c}D\hfill \\ B\hfill \end{array}\right]+\mathrm{rank}\left(C\right).\hfill \end{array}$$

(6) 

Y is re-negative definite if and only if

$$\begin{array}{cc}\hfill i_{-}\left(G_1\right)& =\mathrm{rank}\left[\begin{array}{cc}A\hfill & C\hfill \end{array}\right]+\mathrm{rank}\left(D\right)\mathit{and}{i}_{-}\left(G_2\right)\ge \mathrm{rank}\left[\begin{array}{c}D\hfill \\ B\hfill \end{array}\right]+\mathrm{rank}\left(C\right),\hfill \\ \hfill \mathit{or}{i}_{-}\left(G_1\right)& \ge \mathrm{rank}\left[\begin{array}{cc}A\hfill & C\hfill \end{array}\right]+\mathrm{rank}\left(D\right)\mathit{and}{i}_{-}\left(G_2\right)=\mathrm{rank}\left[\begin{array}{c}D\hfill \\ B\hfill \end{array}\right]+\mathrm{rank}\left(C\right).\hfill \end{array}$$

(7) 

Y is re-nonnegative definite if and only if

$$\begin{array}{cc}\hfill w_1-w_4+i_{-}\left(G_1\right)-w_2& =0\mathit{and}{w}_1-w_4+i_{-}\left(G_2\right)-w_3\le 0,\hfill \\ \hfill \mathit{or}{w}_1-w_4+i_{-}\left(G_1\right)-w_2& \le 0\mathit{and}{w}_1-w_4+i_{-}\left(G_2\right)-w_3=0.\hfill \end{array}$$

(8) 

Y is re-nonpositive definite if and only if

$$\begin{array}{cc}\hfill w_1-w_4+i_{+}\left(G_1\right)-w_2& =0\mathit{and}{w}_1-w_4+i_{+}\left(G_2\right)-w_3\le 0,\hfill \\ \hfill \mathit{or}{w}_1-w_4+i_{+}\left(G_1\right)-w_2& \le 0\mathit{and}{w}_1-w_4+i_{+}\left(G_2\right)-w_3=0.\hfill \end{array}$$

Remark 29. 

When both B and C in Theorem 24 are taken as identity matrices with $k_2=q$ and $m=k_3$, we can immediately obtain the equivalent conditions for the existence of re-positive definite, re-negative definite, re-nonnegative definite, and re-nonpositive definite solutions of Equation (5).

5.8. $\eta$-Hermitian and $\eta$-Skew-Hermitian Solutions

Let $\mathbb{F}=\mathbb{H}$. The $\eta$-Hermitian matrices have been employed in statistical multichannel processing and widely linear modeling (see [116,117]). Yuan and Wang [118] considered the least-squares $\eta$-Hermitian solution with the least norm to the quaternion matrix equation

$$AXE+FXB=C$$

with unknown X. After that, He and Wang [119] investigated the $\eta$-Hermitian solutions of the matrix equation

$$AX{A^{\eta }}^{*}+BY{B^{\eta }}^{*}=C,$$

(46)

where A, B, and C are given quaternion matrices with appropriate orders. Moreover, a solution pair $(X,Y)$ of Equation (46) is called to be $\eta$-Hermitian ($\eta$-skew-Hermitian) if both X and Y are $\eta$-Hermitian ($\eta$-skew-Hermitian).

Theorem 25 

(Corollaries 3.5 and 4.3, [119]). Let A, B, and C be given over $\mathbb{H}$ such that $C={C^{\eta }}^{*}$. Set

$$M=R_{A}B\mathit{and}S=B{L}_M.$$

Then, the following are equivalent:

(1) 

Equation (46) has an η-Hermitian solution pair $(X,Y)$;

(2) 

$R_M{R}_{A}C=0$ and $R_{A}C{\left(R_B\right)}^{\eta *}=0$;

(3) 

$\mathrm{rank}\left[\begin{array}{cc}A& C\\ 0& B^{\eta *}\end{array}\right]=\mathrm{rank}\left(A\right)+\mathrm{rank}\left(B\right)$ and $\mathrm{rank}\left[\begin{array}{ccc}A\hfill & B\hfill & C\hfill \end{array}\right]=\mathrm{rank}\left[\begin{array}{cc}A\hfill & B\hfill \end{array}\right]$.

In this case,

$$\begin{array}{cc}\hfill X=& A^{†}C{\left(A^{†}\right)}^{\eta *}-{\displaystyle \frac{1}{2}}{A}^{†}B{M}^{†}C[I+{\left(B^{†}\right)}^{\eta *}{S}^{\eta *}]{\left(A^{†}\right)}^{\eta *}\hfill \\ \hfill & -{\displaystyle \frac{1}{2}}{A}^{†}(I+S{B}^{†})C{\left(M^{†}\right)}^{\eta *}{B}^{\eta *}{\left(A^{†}\right)}^{\eta *}-A^{†}S{W}_2{S}^{\eta *}{\left(A^{†}\right)}^{\eta *}+L_{A}U+U^{\eta *}{\left(L_A\right)}^{\eta },\hfill \\ \hfill Y=& {\displaystyle \frac{1}{2}}{M}^{†}C{\left(B^{†}\right)}^{\eta *}[I+{\left(S^{†}S\right)}^{\eta }]+{\displaystyle \frac{1}{2}}(I+S^{†}S)B^{†}C{\left(M^{†}\right)}^{\eta *}\hfill \\ & +L_M{W}_2{\left(L_M\right)}^{\eta }+V{L}_B^{\eta }+L_B{V}^{\eta *}+L_M{L}_S{W}_1+W_1^{\eta *}{\left(L_S\right)}^{\eta }{\left(L_M\right)}^{\eta },\hfill \end{array}$$

where $W_1$, U, V, and $W_2=W_2^{\eta *}$ are arbitrary quaternion matrices with appropriate sizes, and

$$\begin{array}{cc}\hfill & \underset{AX{A}^{\eta *}+BY{B}^{\eta *}=C}{min}\mathrm{rank}\left(X\right)=2\mathrm{rank}\left[\begin{array}{cc}C& B\end{array}\right]-\mathrm{rank}\left[\begin{array}{cc}0& B^{\eta *}\\ B& C\end{array}\right],\hfill \\ \hfill & \underset{AX{A}^{\eta *}+BY{B}^{\eta *}=C}{min}\mathrm{rank}\left(Y\right)=2\mathrm{rank}\left[\begin{array}{cc}A& C\end{array}\right]-\mathrm{rank}\left[\begin{array}{cc}0& A^{\eta *}\\ A& C\end{array}\right].\hfill \end{array}$$

Remark 30. 

Inspired by Theorem 25, we now consider the η-Hermitian solutions of Equation (5) over $\mathbb{H}$, namely finding X and Y over $\mathbb{H}$ such that

$$AX+YB=C,X={X^{\eta }}^{*},\mathit{and}Y={Y^{\eta }}^{*},$$

(47)

where A, B, and C are given over $\mathbb{H}$. By means of

$$X={\displaystyle \frac{1}{2}}\left(\widehat{X}+{\widehat{X}}^{\eta *}\right)\mathit{and}Y={\displaystyle \frac{1}{2}}\left(\widehat{Y}+{\widehat{Y}}^{\eta *}\right),$$

it is easy to check that the following are equivalent:

(1) 

The statement (47) holds.

(2) 

There exist the matrices $\widehat{X}$ and $\widehat{Y}$ over $\mathbb{H}$ such that

$$A\widehat{X}+\widehat{Y}B=C\mathit{and}\widehat{X}{A^{\eta }}^{*}+{B^{\eta }}^{*}\widehat{Y}={C^{\eta }}^{*}.$$

(3) 

There exist the matrices $\widehat{X}$ and $\widehat{Y}$ over $\mathbb{H}$ such that

$$\left[\begin{array}{cc}A& I\end{array}\right]\widehat{X}\left[\begin{array}{cc}I& 0\\ 0& {A^{\eta }}^{*}\end{array}\right]+\left[\begin{array}{cc}I& {B^{\eta }}^{*}\end{array}\right]\widehat{Y}\left[\begin{array}{cc}B& 0\\ 0& I\end{array}\right]=\left[\begin{array}{cc}C& {C^{\eta }}^{*}\end{array}\right].$$

Therefore, solving the η-Hermitian solutions of Equation (5) reduces to solving an equation of the form

$$AXB+CYD=E,$$

which has been solved by Baksalary and Kala [120]. By the same method, we have

(1) 

There exists a matrix pair $(X,Y)$ such that

$$AX+YB=C\mathit{and}X={X^{\eta }}^{*},$$

if and only if there exist the matrices $\widehat{X}$, $\widehat{Y}$, and $\widehat{Z}$ such that

$$A\widehat{X}+\widehat{Y}B=C\mathit{and}\widehat{X}{A^{\eta }}^{*}+{B^{\eta }}^{*}\widehat{Z}={C^{\eta }}^{*}.$$

(48)

(2) 

There exists a matrix pair $(X,Y)$ such that

$$AX+YB=C\mathit{and}Y={Y^{\eta }}^{*},$$

if and only if there exist the matrices $\widehat{X}$, $\widehat{Y}$, and $\widehat{Z}$ such that

$$A\widehat{X}+\widehat{Y}B=C\mathit{and}\widehat{Z}{A^{\eta }}^{*}+{B^{\eta }}^{*}\widehat{Y}={C^{\eta }}^{*}.$$

(49)

Moreover, solvability conditions and expressions of the general solution to Equations (48) and (49) can be obtained by Theorem 2.1 of [121].

Kyrchei [122] further discussed the $\eta$-skew-Hermitian solutions of Equation (46) under $C=-{C^{\eta }}^{*}$ using a method similar to Theorem 25.

Theorem 26 

(Corollary 4.4, [122]). Under the hypotheses of Theorem 25, the following are equivalent:

(1) 

Equation (46) has an η-skew-Hermitian solution pair $(X,Y)$;

(2) 

$R_M{R}_{A}C=0$ and $R_{A}C{\left(R_B\right)}^{\eta *}=0$;

(3) 

$\mathrm{rank}\left[\begin{array}{cc}A& C\\ 0& B^{{\eta }^{*}}\end{array}\right]=\mathrm{rank}\left(A\right)+\mathrm{rank}\left(B\right)$ and $\mathrm{rank}\left[\begin{array}{ccc}A\hfill & B\hfill & C\hfill \end{array}\right]=\mathrm{rank}\left[\begin{array}{cc}A\hfill & B\hfill \end{array}\right]$;

in which case, the η-skew-Hermitian solutions of Equation (46) are

$$\begin{array}{cc}\hfill X=& A^{†}C{\left(A^{†}\right)}^{\eta *}-{\displaystyle \frac{1}{2}}\left[A^{†}B{M}^{†}C{\left(A^{†}\right)}^{\eta *}+A^{†}C{\left(A^{†}B{M}^{†}\right)}^{\eta *}\right]\hfill \\ \hfill & -{\displaystyle \frac{1}{2}}\left[A^{†}B{M}^{†}C{\left(A^{†}S{B}^{†}\right)}^{\eta *}+A^{†}S{B}^{†}C{\left(A^{†}B{M}^{†}\right)}^{\eta *}\right]-A^{†}S{W}_2{\left(A^{†}S\right)}^{\eta *}-L_{A}U+{\left(L_{A}U\right)}^{\eta *},\hfill \\ \hfill Y=& {\displaystyle \frac{1}{2}}\left[M^{†}C{\left(B^{†}\right)}^{\eta *}+B^{†}C{\left(M^{†}\right)}^{\eta *}\right]+{\displaystyle \frac{1}{2}}\left[M^{†}C{\left(B^{†}\right)}^{\eta *}{\left(Q_S\right)}^{\eta *}+Q_S{B}^{†}C{\left(M^{†}\right)}^{\eta *}\right]\hfill \\ \hfill & +L_M{W}_2{L}_M^{†}+V{L}_B^{†}-L_B{V}^{\eta *}+L_M{L}_S{W}_1-W_1^{\eta *}{L}_S^{†}{L}_M^{\eta *},\hfill \end{array}$$

where $W_1$, U, V, and $W_2=W_2^{\eta *}$ are arbitrary over $\mathbb{H}$ with appropriate sizes.

Remark 31. 

Using a method similar to that in Remark 30, one can also consider the η-skew-Hermitian solutions of Equation (5) over $\mathbb{H}$, i.e., finding X and Y such that

$$AX+YB=C,X=-{X^{\eta }}^{*},\mathit{and}Y=-{Y^{\eta }}^{*}.$$

Remark 32. 

In terms of the determinant representations of the MP inverse over $\mathbb{H}$ (i.e., Theorem 10), Kyrchei [122] also showed the Cramer’s rules for the partial η-Hermitian and η-skew-Hermitian solution to Equation (46) under $C={C^{\eta }}^{*}$ and $C=-{C^{\eta }}^{*}$, respectively.

5.9. $\varphi$-Hermitian Solutions

Let $\mathbb{F}=\mathbb{H}$. Rodman [7] introduced the quaternion matrix $A_{\varphi }$ over $\mathbb{H}$, a generation of $A^{*}$ and ${A^{\eta }}^{*}$, and defined the $\varphi$-Hermitian quaternion matrix as follows:

Definition 9 

([7]). Let $\varphi :\mathbb{H}\to \mathbb{H}$ be a map.

(1) 

We call ϕ an anti-endomorphism if for any $\alpha ,\beta \in \mathbb{H}$, ϕ satisfies

$$\varphi \left(\alpha \beta \right)=\varphi \left(\beta \right)\varphi \left(\alpha \right)\mathit{and}\varphi (\alpha +\beta )=\varphi \left(\alpha \right)+\varphi \left(\beta \right).$$

An anti-endomorphism ϕ is called an involution if ${\varphi }^2$ is the identity map.

(2) 

Let ϕ be a nonzero involution. Then, ϕ can be represented as a matrix in ${\mathbb{R}}^{4\times 4}$ with respect to the basis $\{1,\mathbf{i},\mathbf{j},\mathbf{k}\}$, i.e.,

$$\varphi =\left[\begin{array}{cc}1& 0\\ 0& T\end{array}\right],$$

where either $T=-I_3$ (in which case ϕ is called a standard involution), or $T\in {\mathbb{R}}^{3\times 3}$ is an orthogonal symmetric matrix with the eigenvalues $\{1,1,-1\}$ (in which case ϕ is called a nonstandard involution).

(3) 

Let ϕ be a nonstandard involution and $A=\left[a_{i,j}\right]\in {\mathbb{H}}^{m\times n}$. Define

$$\varphi \left(A\right)=\left[\varphi \left(a_{i,j}\right)\right]\in {\mathbb{H}}^{m\times n}\mathit{and}{A}_{\varphi }=\varphi \left(A^T\right)\in {\mathbb{H}}^{n\times m}.$$

If $A=A_{\varphi }$ with $m=n$, then A is called a ϕ-Hermitian matrix.

He et al. [123] considered the $\varphi$-Hermitian solution $Z=Z_{\varphi }$ of the following system

$$\left\{\begin{array}{c}{A}_{1}X-Y{B}_1=C_1,\hfill \\ A_{2}Z-Y{B}_2=C_2,\hfill \end{array}\right.$$

(50)

where $A_i$, $B_i$, and $C_i(i=1,2)$ are given matrices over $\mathbb{H}$ with appropriate orders.

Theorem 27 

(Theorem 4.5, [123]). Let

$$\begin{array}{cc}\hfill & A_{11}=R_{B_2}{B}_1,B_{11}=R_{A_2}{A}_2,C_{11}=B_1{L}_{A_{11}},D_{11}=R_{A_1}(R_{A_2}{C}_2{B}_2^{†}{B}_1-C_1)L_{A_{11}},A_{22}=[L_{A_2},-{\left(R_{C_{11}}{B}_2\right)}_{\varphi }],\hfill \\ \hfill & B_{22}=\left[\begin{array}{c}{R}_{C_{11}}{B}_2\\ -{\left(L_{A_2}\right)}_{\varphi }\end{array}\right],C_{22}={\left(A_2^{†}{C}_2{L}_{B_2}\right)}_{\varphi }+{\left(B_2\right)}_{\varphi }{\left(C_{11}^{†}\right)}_{\varphi }{D}_{22}{\left(B_{11}^{†}\right)}_{\varphi }-A_2^{†}{C}_2-B_{11}^{†}{D}_{11}{C}_{11}^{†}{B}_2,\hfill \\ \hfill & A=R_{A_{22}}{L}_{B_{11}},B=B_2{L}_{B_{22}},C=-R_{A_{22}}{\left(B_2\right)}_{\varphi },D={\left(L_{B_{11}}\right)}_{\varphi }{L}_{B_{22}},E=R_{A_{22}}{C}_{22}{L}_{B_{22}},M=R_{A}C,\hfill \end{array}$$

Then, the following are equivalent:

(1) 

The system (50) has a solution $(X,Y,Z)$ such that $Z=Z_{\varphi }$.

(2) 

The following rank equalities hold:

$$\begin{array}{cc}\hfill & \mathrm{rank}\left[\begin{array}{cc}{C}_i& A_i\\ B_i& 0\end{array}\right]=\mathrm{rank}\left(A_i\right)+\mathrm{rank}\left(B_i\right),i=1,2,\hfill \\ \hfill & \mathrm{rank}\left[\begin{array}{cccc}{C}_1& C_2& A_1& A_2\\ B_1& B_2& 0& 0\end{array}\right]=\mathrm{rank}\left[\begin{array}{cc}{A}_1& A_2\end{array}\right]+\mathrm{rank}\left[\begin{array}{cc}{B}_1& B_2\end{array}\right],\hfill \\ \hfill & \mathrm{rank}\left[\begin{array}{cccc}{C}_1& C_2{\left(A_2\right)}_{\varphi }-A_2{\left(C_2\right)}_{\varphi }& A_1& A_2{\left(B_2\right)}_{\varphi }\\ B_1& B_2{\left(A_2\right)}_{\varphi }& 0& 0\end{array}\right]=\mathrm{rank}\left[\begin{array}{cc}{A}_1& A_2{\left(B_2\right)}_{\varphi }\end{array}\right]+\mathrm{rank}\left[\begin{array}{cc}{B}_1& B_2{\left(A_2\right)}_{\varphi })\end{array}\right],\hfill \end{array}$$

$$\begin{array}{cc}\hfill & \mathrm{rank}\left[\begin{array}{cc}{C}_2{\left(A_2\right)}_{\varphi }-A_2{\left(C_2\right)}_{\varphi }& A_2{\left(B_2\right)}_{\varphi }\\ B_2{\left(A_2\right)}_{\varphi }& 0\end{array}\right]=2\mathrm{rank}\left(A_2{\left(B_2\right)}_{\varphi }\right),\hfill \\ \hfill & \mathrm{rank}\left[\begin{array}{cccc}{C}_1& C_2{\left(A_2\right)}_{\varphi }-A_2{\left(C_2\right)}_{\varphi }& A_1& A_2{\left(B_2\right)}_{\varphi }\\ 0& -{\left(C_1\right)}_{\varphi }& 0& {\left(B_1\right)}_{\varphi }\\ B_1& B_2{\left(A_2\right)}_{\varphi }& 0& 0\\ 0& {\left(A_1\right)}_{\varphi }& 0& 0\end{array}\right]=2\mathrm{rank}\left[\begin{array}{cc}{A}_1& A_2{\left(B_2\right)}_{\varphi }\\ 0& {\left(B_1\right)}_{\varphi }\end{array}\right].\hfill \end{array}$$

(3) 

The following equations hold:

$$R_{A_2}{C}_2{L}_{B_2}=0,D_{11}{L}_{C_{11}}=0,R_{B_{11}}{D}_{11}=0,R_M{R}_{A}E=0,R_{C}E{L}_B=0,R_{A}E{L}_D=0.$$

In this case,

$$\begin{array}{cc}\hfill X& ={\displaystyle \frac{1}{2}}(X_1+{\left(X_5\right)}_{\varphi }),Y={\displaystyle \frac{1}{2}}(X_2+{\left(X_4\right)}_{\varphi }),\mathrm{and}Z={\displaystyle \frac{1}{2}}(X_3+{\left(X_3\right)}_{\varphi }),\hfill \end{array}$$

where $X_1,X_2,\dots ,X_5$ are given in (Formulas (4.24)–(4.39), [123]).

Remark 33. 

(1) 

When $A_1=B_1=C_1=0$, Theorem 27 yields the result for

$$A_{2}Z-Y{B}_2=C_2\mathit{subject}\mathit{to}Z=Z_{\varphi },$$

which can be regarded as Equation (1) under the constrain that X is ϕ-Hermitian, i.e.,

$$AX-YB=C\mathit{subject}\mathit{to}X=X_{\varphi }.$$

(51)

(2) 

Note that ϕ-Hermitian matrices are a generalization of Hermitian matrices. In Theorems 5.1 and 5.2 of [121], He and Wang have investigated the following problem over $\mathbb{C}$:

$$\begin{array}{cc}\hfill & AX-YB=C\mathit{subject}\mathit{to}X=X^{*}(\mathit{or}Y=Y^{*}),\hfill \end{array}$$

which is clearly similar to the problem (51).

(3) 

By the same method as in Remark 30, we can also discuss the following problem:

$$AX-YB=C\mathit{subject}\mathit{to}X=X_{\varphi }\mathit{and}Y=Y_{\varphi }.$$

5.10. Equality-Constrained Solutions

Let $\mathbb{F}=\mathbb{C}$. Wang et al. [124] considered the solvability conditions and the general solution for Equation (5) over $\mathbb{C}$ under the following equality constraints:

$$A_{1}X=C_1,Y{B}_2=C_2,A_{3}X{B}_3=C_3,\mathrm{and}{A}_{4}Y{B}_4=C_4,$$

(52)

where $A_1$, $A_3$, $A_4$, $B_2$, $B_3$, $B_4$, $C_1$, $C_2$, $C_3$, and $C_4$ are given.

Theorem 28 

(Theorem 3.2, [124]). Let $A_1$, $B_1$, $C_1$, $B_2$, $C_2$, $A_3$, $B_3$, $C_3$, $A_4$, $B_4$, $C_4$, A, B, and C be given matrices over $\mathbb{C}$ with appropriate sizes. Set

$$\begin{array}{cc}\hfill & T=A_3{L}_{A_1},K=R_{B_2}{B}_4,{\varphi }_1=A_1^{†}{C}_1+L_{A_1}{T}^{†}(C_3-A_3{A}_1^{†}{C}_1{B}_3)B_3^{†},\hfill \\ \hfill & {\varphi }_2=C_2{B}_2^{†}+A_4^{†}(C_4-A_4{C}_2{B}_2^{†}{B}_4)K^{†}{R}_{B_2},A_{11}=A{L}_{A_1}{L}_T,B_{11}=R_K{R}_{B_2}B,\hfill \\ \hfill & C_{33}=A{L}_{A_1},D_{33}=R_{B_3},C_{44}=L_{A_4},D_{44}=R_{B_2}B,E_{11}=C-A{\varphi }_1-{\varphi }_{2}B,\hfill \\ \hfill & A_a=R_{A_{11}}{C}_{33},B_b=D_{33}{L}_{B_{11}},C_c=R_{A_a}{C}_{44},D_d=D_{44}{L}_{B_{11}},\hfill \\ \hfill & E=R_{A_{11}}{E}_{11}{L}_{B_{11}},M=R_{A_a}{C}_c,N=D_d{L}_{B_b},S=C_c{L}_M.\hfill \end{array}$$

Then, the following are equivalent:

(1) 

Equation (5) under the constraints (52) is consistent.

(2) 

The following rank equations hold:

$$\mathrm{rank}[A_1,C_1]=\mathrm{rank}\left(A_1\right),\mathrm{rank}\left[\begin{array}{c}{C}_3\\ B_3\end{array}\right]=\mathrm{rank}\left(B_3\right),\mathrm{rank}\left[\begin{array}{cc}{A}_1& C_1{B}_3\\ A_3& C_3\end{array}\right]=\mathrm{rank}\left[\begin{array}{c}{A}_1\\ A_3\end{array}\right],$$

$$\mathrm{rank}\left[\begin{array}{c}{C}_2\\ B_2\end{array}\right]=\mathrm{rank}\left(B_2\right),\mathrm{rank}[A_4,C_4]=\mathrm{rank}\left(A_4\right),\mathrm{rank}\left[\begin{array}{cc}{C}_4& A_4{C}_2\\ B_4& B_2\end{array}\right]=\mathrm{rank}[B_4,B_2],$$

$$\mathrm{rank}\left[\begin{array}{ccc}0& B_2& B{B}_3\\ A& C_2& C{B}_3\\ A_3& 0& C_3\\ A_1& 0& C_1{B}_3\end{array}\right]=\mathrm{rank}\left[\begin{array}{ccc}0& B_2& B{B}_3\\ A& 0& 0\\ A_3& 0& 0\\ A_1& 0& 0\end{array}\right],\mathrm{rank}\left[\begin{array}{ccc}0& B& B_2\\ A& C& C_2\\ A_1& C_1& 0\end{array}\right]=\mathrm{rank}\left[\begin{array}{ccc}0& B& B_2\\ A& 0& 0\\ A_1& 0& 0\end{array}\right],$$

$$\begin{array}{cc}\hfill & \mathrm{rank}\left[\begin{array}{cccc}0& B& B_4& B_2\\ A_{4}A& A_{4}C& C_4& A_4{C}_2\\ A_1& C_1& 0& 0\end{array}\right]=\mathrm{rank}\left[\begin{array}{cccc}0& B& B_4& B_2\\ A_{4}A& 0& 0& 0\\ A_1& 0& 0& 0\end{array}\right],\hfill \\ \hfill & \mathrm{rank}\left[\begin{array}{cccc}0& B_4& B_2& B{B}_3\\ A_3& 0& 0& C_3\\ A_1& 0& 0& C_1{B}_3\\ A_{4}A& C_4& A_4{C}_2& A_{4}C{B}_3\end{array}\right]=\mathrm{rank}\left[\begin{array}{cccc}0& B_4& B_2& B{B}_3\\ A_3& 0& 0& 0\\ A_1& 0& 0& 0\\ A_{4}A& 0& 0& 0\end{array}\right].\hfill \end{array}$$

(3) 

The following equations hold:

$$\begin{array}{cc}\hfill & R_{A_1}{C}_1=0,R_T(C_3-A_3{A}_1^{†}{C}_1{B}_3)=0,C_3{L}_{B_3}=0,\hfill \\ \hfill & C_2{L}_{B_2}=0,R_{A_4}{C}_4=0,(C_4-A_4{C}_2{B}_2^{†}{B}_4)L_K=0,\hfill \\ \hfill & R_M{R}_{A_a}E=0,E{L}_{B_b}{L}_N=0,R_{A_a}E{L}_{D_d}=0,R_{C_c}E{L}_{B_b}=0.\hfill \end{array}$$

In this case,

$$\begin{array}{cc}\hfill X& =A_1^{†}{C}_1+L_{A_1}{T}^{†}(C_3-A_3{A}_1^{†}{C}_1{B}_3)B_3^{†}+L_{A_1}{L}_T{Z}_1+L_{A_1}{W}_1{R}_{B_3},\hfill \\ \hfill Y& =C_2{B}_2^{†}+A_4^{†}(C_4-A_4{C}_2{B}_2^{†}{B}_4)K^{†}{R}_{B_2}+L_{A_4}{W}_2{R}_{B_2}+Z_2{R}_K{R}_{B_2},\hfill \\ \hfill Z_1& =A_{11}^{†}(E_{11}-C_{33}{W}_1{D}_{33}-C_{44}{W}_2{D}_{44})-A_{11}^{†}{V}_7{B}_1+L_{A_{11}}{V}_6,\hfill \\ \hfill Z_2& =R_{A_{11}}(E_{11}-C_{32}{W}_1{D}_{33}-C_{44}{W}_2{D}_{44})B_{11}^{†}+A_{11}{A}_{11}^{†}{V}_7+V_8{R}_{B_{11}},\hfill \\ \hfill W_1& =A_a^{†}E{B}_b^{†}-A_a^{†}{C}_c{M}^{†}E{B}_b^{†}-A_a^{†}S{C}_c^{†}E{N}^{†}{D}_d{B}_b^{†}-A_a^{†}S{V}_4{R}_N{D}_d{B}_b^{†}+L_{A_a}{V}_1+V_2{R}_{B_b},\hfill \\ \hfill W_2& =M^{†}E{D}_d^{†}+S^{†}S{C}_c^{†}E{N}^{†}+L_M{L}_S{V}_3+L_M{V}_4{R}_N+V_5{R}_{D_d},\hfill \end{array}$$

where $V_1,\dots ,V_8$ are arbitrary matrices over $\mathbb{C}$ with appropriate orders.

Remark 34. 

Inspired by Theorem 28, using the Kronecker product and the vectorization operation, Wang et al. [125] investigated the minimum-norm least-squares solution for the quaternion tensor system under the Einstein product:

$$\begin{array}{cc}\hfill & \mathcal{A}{*}_N\mathcal{X}+\mathcal{Y}{*}_N\mathcal{B}=\mathcal{C}\hfill \\ \hfill \mathit{subject}\mathit{to}& \left\{\begin{array}{c}{\mathcal{A}}_1{*}_N\mathcal{X}={\mathcal{C}}_1,{\mathcal{A}}_3{*}_N\mathcal{X}{*}_N{\mathcal{B}}_3={\mathcal{C}}_3,\hfill \\ \mathcal{Y}{*}_N{\mathcal{B}}_2={\mathcal{C}}_2,{\mathcal{A}}_4{*}_N\mathcal{Y}{*}_N{\mathcal{B}}_4={\mathcal{C}}_4,\hfill \end{array}\right.\hfill \end{array}$$

(53)

where $\mathcal{X}$ and $\mathcal{Y}$ are unknown tensors and the others are given tensors over $\mathbb{H}$. Thus, the minimum-norm least-squares solution of the tensor Equation (53) is directly given in (Corollary 3.4, [125]), which is also expounded in Section 6.5.

6. Various Generalizations of GSE

This section shows the generalizations of GSE to diverse domains such as various rings, dual numbers, dual quaternions, linear operators, tensors, and matrix polynomials, as well as its more general forms. This embodies another enchantment of mathematics: constantly exploring more general problems to ultimately reveal the most essential conclusions.

6.1. Generalizing RET over Different Rings

RET given in Section 3 characterizes the equivalent condition for the solvability of GSE over a field. This subsection mainly generalizes RET over the following algebraic structures: unit regular rings, principal ideal rings, commutative rings, division rings, Artinian rings, etc. To simplify expressions, we first introduce Guralnick’s definition in [126].

Definition 10 

([126]). If (3) is equivalent to (4) over a ring $\mathbb{F}$, then we say that $\mathbb{F}$ has the equivalence property.

6.1.1. Generalizing RET over Unit Regular Rings

A ring $\mathbb{F}$ is called to be unit regular if for any $a\in \mathbb{F}$, there exists a unit $u\in \mathbb{F}$ such that

$$a=aua.$$

Hartwig [127] generalized RET to a unit regular ring, also extending Theorems 3 and 4.

Theorem 29 

([127]). Let $\mathbb{F}$ be a unit regular ring, and

$$M=\left[\begin{array}{cc}a& c\\ 0& b\end{array}\right]\in {\mathbb{F}}^{2\times 2}.$$

Then, the following statements are equivalent:

(1) 

M has an inner inverse with the form of $\left[\begin{array}{cc}r& s\\ 0& t\end{array}\right]\in {\mathbb{F}}^{2\times 2}$;

(2) 

$ax-yb=c$ has a solution pair $x,y\in \mathbb{F}$;

(3) 

$(1-a{a}^{-})c(1-b^{-}b)=0$ for all $a^{-}$ and $b^{-}$;

(4) 

$a_{1}c{b}_1=0$, where $a_1\in \{x\in \mathbb{F}\mid xa=0\}$ and $b_1\in \{x\in \mathbb{F}\mid bx=0\}$;

(5) 

$M=p\left[\begin{array}{cc}a& 0\\ 0& b\end{array}\right]q$, where $p,q\in \mathbb{F}$ are invertible;

(6) 

$(1-a{a}^{(1,2)})c(1-b^{(1,2)}b)=0$ for all $a^{(1,2)}$ and $b^{(1,2)}$;

(7) 

$M^{(1,2)}=\left[\begin{array}{cc}{a}^{(1,2)}& -a^{(1,2)}c{b}^{(1,2)}\\ 0& b^{(1,2)}\end{array}\right]$ is a reflexive inverse of M.

If $\mathbb{F}$ is a skewfield (or a commutative ring without zero divisors) and $a,c,b\in {\mathbb{F}}^{n\times n}$, then items (1)–(7) are also equivalent to

(5a) 

$\mathrm{rank}\left(M\right)=\mathrm{rank}\left(a\right)+\mathrm{rank}\left(b\right)$;

(5b) 

$\mathrm{rank}\left((1-a{a}^{(1,2)})c(1-b^{(1,2)}b)\right)=0$.

Remark 35. 

In the conclusions of [127], Hartwig mentioned that RET also holds for matrices over Euclidean domains and unit regular rings. These rings are finite and elementary divisor rings satisfying the cancellation law. However, whether these properties are sufficient to ensure that the validity of RET remains an open problem. In [126], Guralnick considered parts of this problem.

6.1.2. Generalizing RET over Principal Ideal Domains

Building on Theorem 2 of [128], Feinberg [129] considered RET over principal ideal domains and further extended it to a more general form.

Theorem 30 

(Theorems 1 and 2, [129]). Let $\mathbb{F}$ be a principal ideal domain.

(1) 

Let $A\in {\mathbb{F}}^{r\times r}$, $B\in {\mathbb{F}}^{s\times s}$, and $C\in {\mathbb{F}}^{r\times s}$. Then, the matrix equation

$$AX+YB=C$$

is consistent if and only if

$$\left[\begin{array}{cc}A& C\\ 0& B\end{array}\right]\mathit{and}\left[\begin{array}{cc}A& 0\\ 0& B\end{array}\right]$$

are equivalent.

(2) 

Let $M_{ij}\in {\mathbb{F}}^{r_j\times r_j}$ for $1\le i\le j\le t$. Then,

$$\left[\begin{array}{cccc}{M}_{11}& M_{12}& \dots & M_{1t}\\ 0& M_{22}& \dots & M_{2t}\\ ⋮& ⋮& \ddots & ⋮\\ 0& \dots & 0& M_{tt}\end{array}\right]\mathit{and}\left[\begin{array}{cccc}{M}_{11}& 0& \dots & 0\\ 0& M_{22}& \dots & 0\\ ⋮& ⋮& \ddots & ⋮\\ 0& \dots & 0& M_{tt}\end{array}\right]$$

are equivalent if and only if there exist $X_{ij},Y_{ij}\in {\mathbb{F}}^{r_i\times r_j}$ such that

$$M_{ij}=M_{ii}{X}_{ij}+\sum _{k=i+1}^j{Y}_{ik}{M}_{kj}$$

for $1\le i<j\le t$.

Remark 36. 

Feinberg [129] also noted that it is easy to generalize Theorem 1 to a principal ideal domain by a method similar to that of Theorem 1 of [129].

Let $\mathbb{F}=\mathbb{C}$. In view of Remark 6, we can see that (4) implies that there exist the equivalence matrices to be of the upper block triangular, i.e.,

$$\left[\begin{array}{cc}I& -Y\\ 0& I\end{array}\right]\mathrm{and}\left[\begin{array}{cc}I& X\\ 0& I\end{array}\right].$$

Olshevsky [22] generalized the above conclusion by showing that, if any block upper triangular matrix is equivalent to its block diagonal part, then the equivalent matrices can also be chosen in the form of a block upper triangular matrix.

Theorem 31 

(Theorem 3.2, [22]). Let $A_{ij}\in {\mathbb{C}}^{n_i\times n_j}$ for $1\le i\le j\le k$. If

$$G=\left[\begin{array}{ccccc}{A}_{11}& 0& \dots & \dots & 0\\ 0& A_{22}& 0& \dots & 0\\ ⋮& 0& \ddots & \ddots & ⋮\\ ⋮& \ddots & \ddots & 0\\ 0& \dots & \dots & 0& A_{kk}\end{array}\right]\mathit{and}H=\left[\begin{array}{ccccc}{A}_{11}& A_{12}& \dots & \dots & A_{1k}\\ 0& A_{22}& A_{23}& \dots & A_{2k}\\ ⋮& 0& \ddots & & ⋮\\ ⋮& \ddots & \ddots & \ddots & ⋮\\ 0& \dots & \dots & 0& A_{kk}\end{array}\right]$$

are equivalent, then there exist $X_{ij},Y_{ij}\in {\mathbb{C}}^{n_i\times n_j}$ ($1\le i<j\le k$) such that

$$\left[\begin{array}{ccccc}{I}_{n_1}& Y_{12}& \dots & \dots & Y_{1k}\\ 0& I_{n_2}& Y_{23}& \dots & Y_{2k}\\ ⋮& 0& \ddots & & ⋮\\ ⋮& & \ddots & \ddots & ⋮\\ 0& \dots & \dots & 0& I_{n_k}\end{array}\right]G\left[\begin{array}{ccccc}{I}_{n_1}& X_{12}& \dots & \dots & X_{1k}\\ 0& I_{n_2}& X_{23}& \dots & X_{2k}\\ ⋮& 0& \ddots & & ⋮\\ ⋮& & \ddots & \ddots & ⋮\\ 0& \dots & \dots & 0& I_{n_k}\end{array}\right]=H.$$

Remark 37. 

Theorem 31 can also be derived from Theorem 30 (30).

6.1.3. Generalizing RET over Division and Module-Finite Rings

Let $\mathbb{F}$ be a ring with identity. For $G\in {\mathbb{F}}^{a\times b}$ and $H\in {\mathbb{F}}^{c\times d}$, let

$$E_R(G,H)=\left\{(T,S)\mid T\in {\mathbb{F}}^{c\times a},S\in {\mathbb{F}}^{d\times b}\mathrm{and}TG=HS\right\}.$$

Denote

$$M=\left[\begin{array}{cc}A& C\\ 0& B\end{array}\right],$$

where A, B, and C are given in Equation (1). For $(T,S)\in E_R(M,A)$, denote

$$T=\left[\begin{array}{cc}{T}_1& T_2\end{array}\right]\mathrm{and}S=\left[\begin{array}{cc}{S}_1& S_2\end{array}\right],$$

where $T_1\in {\mathbb{F}}^{m\times m}$, $T_2\in {\mathbb{F}}^{m\times s}$, $S_1\in {\mathbb{F}}^{r\times r}$, and $S_2\in {\mathbb{F}}^{r\times n}$. Then,

$$TM=AS\iff \left\{\begin{array}{c}{T}_{1}A=A{S}_1,\hfill \\ T_{1}C+T_{2}B=A{S}_2.\hfill \end{array}\right.$$

Define a map

$$g_{M,R}:E_R(M,A)\to E_R(A,A)\mathrm{by}g((T_1,T_2),(S_1,S_2))=(T_1,S_1).$$

(54)

Gustafson et al. [130] gave a general characterization of the solvability of Equation (1) over $\mathbb{F}$ based on the map $g_{M,R}$, which is essentially similar to the method used by Hartwig in [127].

Theorem 32 

(Lemma 1, [130]). Equation (1) is consistent if and only if the map $g_{M,R}$ defined in (54) is surjective.

Using Theorem 32, Gustafson et al. [130] further proved that RET also holds for a division ring and for a ring which is finitely generated as modules over its center.

Theorem 33 

([130]). A division ring (or a ring that is module-finite over its center) has the equivalence property.

6.1.4. Generalizing RET over Commutative Rings

Let $\mathbb{F}$ be a commutative ring with identity. Gustafson [131] showed that RET remain valid over the commutative ring $\mathbb{F}$. By reducing this problem to Artinian rings, Gustafson employed a simple argument with the composition length. This approach parallels that of Flanders and Wimmer [11], who used linear transformations and subspace dimensions to discuss RET over fields.

Theorem 34 

(Theorem 1, [131]). A commutative ring with identity has the equivalence property.

Guralnick [132] further generalized Theorem 34 to finite sets of matrices over the commutative ring $\mathbb{F}$. Let $\mathbb{F}[x_1,x_2,\dots ,x_t]$ be the polynomial ring over $\mathbb{F}$, where $x_i$$(i=1,\dots ,t)$ is commute with each other and any element in $\mathbb{F}$. For

$$\tilde{C}=\left\{C_i\in {\mathbb{F}}^{m\times n}\mid 1\le i\le r\right\}\mathrm{and}\tilde{D}=\left\{D_i\in {\mathbb{F}}^{m\times n}\mid 1\le i\le r\right\},$$

we call that $\tilde{C}$ and $\tilde{D}$ are simultaneously equivalent if there exist invertible matrices $U\in {\mathbb{F}}^{m\times m}$ and $V\in {\mathbb{F}}^{n\times n}$ such that $U{C}_{i}V=D_i$ for any $1\le i\le r$ (see [132,133]).

Theorem 35 

(Theorem B(i), [132]). For $A_i\in {\mathbb{F}}^{m\times r}$, $B_i\in {\mathbb{F}}^{s\times n}$, and $C_i\in {\mathbb{F}}^{m\times n}$, let

$$M_i=\left[\begin{array}{cc}{A}_i& C_i\\ 0& B_i\end{array}\right]\mathit{and}{N}_i=\left[\begin{array}{cc}{A}_i& 0\\ 0& B_i\end{array}\right],$$

where $1\le i\le t$. Suppose that the polynomial ring $\mathbb{F}[x_1,x_2,\dots ,x_t]$ has the equivalence property. Then, the system of matrix equations

$$\left\{\begin{array}{c}{A}_{1}X-Y{B}_1=C_1,\hfill \\ A_{2}X-Y{B}_2=C_2,\hfill \\ \dots \hfill \\ A_{r}X-Y{B}_r=C_r,\hfill \end{array}\right.$$

(55)

has a common solution pair $X\in {\mathbb{F}}^{r\times n}$ and $Y\in {\mathbb{F}}^{m\times s}$ if and only if

$$\tilde{M}=\left\{M_i\mid 1\le i\le r\right\}\mathit{and}\tilde{N}=\left\{N_i\mid 1\le i\le r\right\}$$

are simultaneously equivalent.

Remark 38. 

Dmytryshyn and their collaborators [12,13] have done significant contributions on the further generalizations of the system (55). Moreover, Dmytryshyn was awarded the SIAM Student Paper Prize 2015 for their work [12].

Let $\mathbb{F}$ be a field with $\mathrm{char}\left(\mathbb{F}\right)\ne 2$. We stipulate that, in the following systems from [12,13], except for the unknown matrices, the remaining matrices are given with appropriate orders over $\mathbb{F}$. In Theorem 4.1 of [12], Dmytryshyn et al. first study the system

$$A_i{X}_k\pm X_j{B}_i=C_i,$$

(56)

where $i=1,\dots ,n,k,j\in \{1,\dots ,m\},1\le m\le 2n,$ and $X_1,\dots ,X_m$ are unknown. Their method for solving the system (56) extends that used in [11]. Based on the system (56), they then considered the main research subject of [12], i.e.,

$$\left\{\begin{array}{c}{A}_i{X}_k\pm X_j{B}_i=C_i,i=1,\dots ,n_1,\hfill \\ F_{i^{\prime }}{X}_{k^{\prime }}\pm X_{j^{\prime }}^{★}{G}_{i^{\prime }}=H_{i^{\prime }},i^{\prime }=1,\dots ,n_2,\hfill \end{array}\right.$$

(57)

where

(i) 

$k,j,k^{\prime },j^{\prime }\in \{1,\dots ,m\}$, $1\le m\le (2n_1+2n_2)$, and $X_1,\dots ,X_m$ are unknown;

(ii) 

For $1\le l\le m$, the symbol $X_l^{★}$ denotes the matrix transpose $X_l^T$ and for the complex number field, also the matrix conjugate transpose $X_l^{*}$,

(see Theorem 1.1, [12]). In Theorem 6.1 of [12], they further generalized the system (57) to the following form:

$$\left\{\begin{array}{c}{A}_i{X}_k{K}_i-L_i{X}_j{B}_i=C_i,i=1,\dots ,n_1,\hfill \\ F_{i^{\prime }}{X}_{k^{\prime }}{M}_{i^{\prime }}+N_{i^{\prime }}{X}_{j^{\prime }}^{★}{G}_{i^{\prime }}=H_{i^{\prime }},i^{\prime }=1,\dots ,n_2,\hfill \end{array}\right.$$

(58)

where $j,k,j^{\prime },k^{\prime }\in \{1,\dots ,m\}$, $1\le m\le (2n_1+2n_2)$, and $X_1,X_2,\dots ,X_m$ are unknown.

Let $\mathbb{F}$ be a skew field of $\mathrm{char}\left(\mathbb{F}\right)\ne 2$ that is finite dimensional over its center. Interestingly, two years later, Dmytryshyn et al. [13] generalized the system (58) over $\mathbb{F}$ including the complex conjugation of unknown matrices, i.e.,

$$A_i{X}_{i^{\prime }}^{{\epsilon }_i}{M}_i-N_i{X}_{i^{\prime \prime }}^{{\delta }_i}{B}_i=C_i,$$

(i) 

Of complex matrix equations, in which ${\epsilon }_i,{\delta }_i\in \{1,\mathrm{C},\mathrm{T},*\}$ and $X^{\mathrm{C}}=\overline{X}$ is the complex conjugate of X;

(ii) 

Of quaternion matrix equations, in which ${\epsilon }_i,{\delta }_i\in \{1,*\}$ and $X^{*}$ is the quaternion conjugate transpose of X,

where $i^{\prime },i^{\prime \prime }\in \{1,\dots ,t\}$, $i=1,\dots ,s$, $1\le t\le 2s$, and $X_1,\dots ,X_t$ are unknown (see Theorem 2, [13]). The system (57) is also extended over $\mathbb{F}$ (see Theorem 1, [13]).

6.1.5. Generalizing RET over Artinian and Noncommutative Rings

Let N be a subgroup of a finitely generated Abelian group M. If M and $N\oplus M/N$ are isomorphic, then N is a direct summand of M? This problem is raised by H. Matsumura, and it is subsequently answered affirmatively by H. Toda. Miyata [134] showed that this result can also be generalized to the more general case of module, i.e., if $\mathbb{F}$ is a commutative Noetherian ring and M is a finitely presented $\mathbb{F}$-module with a submodule N, then

$$M\cong N\oplus M/N\iff N\mathrm{is}\mathrm{a}\mathrm{summand}\mathrm{of}M.$$

(59)

We say that $\mathbb{F}$ has the extension property if (59) holds.

Guralnick [126] proved the equivalence between the equivalence property and the extension property for an Artinian ring $\mathbb{F}$.

Theorem 36 

(Corollary 2.7, [126]). Let $\mathbb{F}$ be a right Artinian ring. Then, $\mathbb{F}$ has the extension property if and only if $\mathbb{F}$ has the equivalence property.

Remark 39. 

In the proof of Theorem 3.4 of [126], Guralnick proposed a new perspective to prove Theorem 34. Moreover, Guralnick in Theorem 3.5 of [126] showed that a commutative ring has the extension property. Differing from Miyata [134] and Gustafson [131], this proof avoids the completion of a local Noetherian ring.

Guralnick [126] found that two special classes of Artinian rings (i.e., semisimple Artinian rings and Artinian principal ideal rings) possess the equivalence property.

Theorem 37 

(Theorem 2.4 and Corollary 4.6, [126]).  

(1) 

A semisimple Artinian ring has the equivalence property.

(2) 

An Artinian principal ideal ring has the equivalence property.

Guralnick [126] finally discussed the more general case where $\mathbb{F}$ is a regular ring, which generalizes Theorem 37 as well items (2) and (5) in Theorem 29.

Theorem 38 

(Theorem 4.3, [126]). Let $\mathbb{F}$ be a regular ring. Then, $\mathbb{F}$ has the equivalence property if and only if ${\mathbb{F}}^{n\times n}$ is directly finite for all n.

Interestingly, Guralnick [135] gave a generalized definition of the equivalence property, i.e., the generalized equivalence property.

Definition 11 

([135]). Let $\mathbb{F}$ be a ring with identity. For $A_{ij}\in {\mathbb{F}}^{n_i\times m_j}$ ($1\le i\le j\le k$), denote

$$\tilde{A}=\left[\begin{array}{ccc}{A}_{11}& & 0\\ & \ddots & \\ 0& & A_{kk}\end{array}\right]\mathrm{and}\tilde{B}=\left[\begin{array}{ccc}{A}_{11}& & A_{ij}\\ & \ddots & \\ 0& & A_{kk}\end{array}\right].$$

We define that $\mathbb{F}$ has the generalized equivalence property if $\tilde{A}$ and $\tilde{B}$ are and equivalent imply that there exist $X_{ij}\in {\mathbb{F}}^{n_i\times n_j}$ and $Y_{ij}\in {\mathbb{F}}^{m_i\times m_j}$ such that

$$\left[\begin{array}{ccc}I& & X_{ij}\\ & \ddots & \\ 0& & I\end{array}\right]\tilde{A}=\tilde{B}\left[\begin{array}{ccc}I& & Y_{ij}\\ & \ddots & \\ 0& & I\end{array}\right].$$

Guralnick [135] then proved that not only semisimple Artinian rings and Artinian principal ideal rings but also module finite R-algebras for a commutative ring R possess the generalized equivalence property. This evidently generalizes Theorem 37.

Theorem 39 

(Theorems 3.3, 3.6 and 3.7, [135]). A semisimple Artinian ring, an Artinian principal ideal ring, or a module finite R-algebra for a commutative ring R has the generalized equivalence property.

Remark 40. 

It is worth noting that the generalized form of Equation (5), i.e.,

$$AXB+CYD=E$$

with unknown X and Y, has also been discussed over fields, principal ideal domains, simple Artinian rings, regular rings with identity, and the associative rings with unit by [120,136,137,138,139], respectively.

6.2. Generalizing RET to a Rank Minimization Problem

In a brief three-page article [140], Lin and Wimmer revealed that RET is essentially a special case of a rank minimization problem over a field $\mathbb{F}$. Let $\mathrm{GI}\left(k\right)$ be the set of all invertible matrices of order k.

Theorem 40 

(Theorem 2, [140]). Let $\mathbb{F}$ be a field, and let $A\in {\mathbb{F}}^{m\times m}$, $B\in {\mathbb{F}}^{n\times n}$, and $C\in {\mathbb{F}}^{m\times n}$. Then,

$$\begin{array}{cc}\hfill & min\left\{\mathrm{rank}\left(AX-YB-C\right)\mid X,Y\in {\mathbb{F}}^{m\times n}\right\}\hfill \\ \hfill =& min\left\{\mathrm{rank}\left(P\left[\begin{array}{cc}A& C\\ 0& B\end{array}\right]-\left[\begin{array}{cc}A& 0\\ 0& B\end{array}\right]Q\right)\mid P,Q\in \mathrm{GI}(m+n)\right\}.\hfill \end{array}$$

Subsequently, Ito and Wimmer [141] generalized Theorem 40 to Bezout domains under the condition that A and B are regular. An integral domain $\mathbb{F}$ with identity is called a Bezout domain if it satisfies that all finitely generated ideals are principal.

Theorem 41 

(Theorem 3.1, [141]). Let $\mathbb{F}$ be a Bezout domain, and let $A\in {\mathbb{F}}^{m\times m}$, $B\in {\mathbb{F}}^{n\times n}$, and $C\in {\mathbb{F}}^{m\times n}$. If A and B are regular over $\mathbb{F}$, then

$$\begin{array}{cc}\hfill & min\left\{\mathrm{rank}\left(AX-YB-C\right)\mid X,Y\in {\mathbb{F}}^{m\times n}\right\}\hfill \\ \hfill =& min\left\{\mathrm{rank}\left(P\left[\begin{array}{cc}A& C\\ 0& B\end{array}\right]-\left[\begin{array}{cc}A& 0\\ 0& B\end{array}\right]Q\right)\mid P,Q\in \mathrm{GI}(m+n)\right\}\hfill \\ \hfill =& \mathrm{rank}\left[\begin{array}{cc}A& C\\ 0& B\end{array}\right]-\mathrm{rank}\left[\begin{array}{cc}A& 0\\ 0& B\end{array}\right]\hfill \\ \hfill =& \dim \left(\mathcal{R}\left(A\right)+C\mathcal{N}\left(B\right)\right)-\mathrm{rank}\left(A\right).\hfill \end{array}$$

Remark 41. 

Theorem 41 also yields the equivalence of (3), (4), and (7) over a Bezout domain directly (see (Corollary 3.2, [141])).

6.3. GSE over Dual Numbers and Dual Quaternions

In 1843, the Irish mathematician William Rowan Hamilton [142] invented quaternions $\mathbb{H}$ (also called Hamilton quaternions or real quaternions). The set $\mathbb{H}$ is a noncommutative associative division algebra, and it also generalizes the real field $\mathbb{R}$ and the complex field $\mathbb{C}$. The quaternion algebra has been effectively applied to mechanics, optics, color image processing, signal processing, computer graphics, flight mechanics, quantum physics, and so on (see [7,47,143,144,145,146,147]).

On the other hand, the British mathematician William Kingdon Clifford [148] invented dual numbers and dual quaternions in 1873. Up to now, dual numbers have been widely used in fields such as kinematics, statics, dynamics, robotics, and brain dynamics (see [149,150,151,152,153,154]).

Definition 12 

([148]). A dual number is defined as

$$\widehat{a}=a_0+a_1\epsilon ,$$

where $a_0,a_1\in \mathbb{R}$, and ε is the dual unit such that

$$\epsilon \ne 0,0\epsilon =\epsilon 0=0,1\epsilon =\epsilon 1=\epsilon \mathit{and}{\epsilon }^2=0.$$

(60)

In this case, $a_0$ is called primal/real/standard part of $\widehat{a}$, and $a_1$ is called dual/infinitesimal part of $\widehat{a}$. The set of all dual numbers is denoted by $\mathbb{D}$, which is a commutative ring.

Fan et al. [155] established solvability conditions and expressions of the general solution to Equation (5) over $\mathbb{D}$ by using the MP inverse and SVD.

Theorem 42 

(Theorem 3, [155]). Let

$$A=A_0+A_1\epsilon \in {\mathbb{D}}^{m\times r},B=B_0+B_1\epsilon \in {\mathbb{D}}^{s\times n},C=C_0+C_1\epsilon \in {\mathbb{D}}^{m\times n},$$

where $A_i\in {\mathbb{R}}^{m\times r}$, $B_i\in {\mathbb{R}}^{s\times n}$, and $C_i\in {\mathbb{R}}^{m\times n}$$(i=0,1)$. The SVDs of $A_0$ and $B_0$ are given by

$$A_0=P\left[\begin{array}{cc}\Omega & 0\\ 0& 0\end{array}\right]Q^T\mathit{and}{B}_0=M\left[\begin{array}{cc}\mathsf{\Lambda }& 0\\ 0& 0\end{array}\right]N^T,$$

where $\Omega =\mathrm{diag}({\omega }_1,\dots ,{\omega }_l)$, $l=\mathrm{rank}\left(A_0\right)$, $\mathsf{\Lambda }=\mathrm{diag}({\lambda }_1,\dots ,{\lambda }_t)$, $t=\mathrm{rank}\left(B_0\right)$, and $P\in {\mathbb{R}}^{m\times m}$, $Q\in {\mathbb{R}}^{r\times r}$, $M\in {\mathbb{R}}^{s\times s}$, and $N\in {\mathbb{R}}^{n\times n}$ are orthogonal. Let

$$P=\left[\begin{array}{cc}{P}_1& P_2\end{array}\right],Q=\left[\begin{array}{cc}{Q}_1& Q_2\end{array}\right],M=\left[\begin{array}{cc}{M}_1& M_2\end{array}\right],N=\left[\begin{array}{cc}{N}_1& N_2\end{array}\right],$$

where $P_2\in {\mathbb{R}}^{m\times (m-l)}$, $Q_2\in {\mathbb{R}}^{r\times (r-l)}$, $M_2\in {\mathbb{R}}^{s\times (s-t)}$, and $N_2\in {\mathbb{R}}^{n\times (n-t)}$. Denote

$$J=C_1-A_1{A}_0^{†}{C}_0-R_{A_0}{C}_0{B}_0^{†}{B}_1,K_1=P_2^T{A}_1{Q}_2,K_2=M_2^T{B}_1{N}_2.$$

Then, Equation (5) has a solution pair $X\in {\mathbb{D}}^{r\times n}$ and $Y\in {\mathbb{D}}^{m\times s}$ if and only if

$$R_{A_0}{C}_0{L}_{B_0}=0\mathit{and}{R}_{K_1}{P}_2^{T}J{N}_2{L}_{K_2}=0,$$

in which case,

$$X=X_0+X_1\epsilon \mathit{and}Y=Y_0+Y_1\epsilon$$

with

$$\begin{array}{cc}\hfill X_0=& A_0^{†}{C}_0+Q_2{K}_1^{†}{P}_2^{T}J{L}_{B_0}-A_0^{†}{P}_1{V}_{11}{M}_1^T{B}_0+Q_2{V}_{23}{N}_1^T+Q_2(L_{K_1}{W}_4-K_1^{†}{W}_3{K}_2)N_2^T,\hfill \\ \hfill Y_0=& R_{A_0}{C}_0{B}_0^{†}+P_2{R}_{K_1}{P}_2^{T}J{N}_2{K}_2^{†}{M}_2^T+P_1(V_{11}{M}_1^T+V_{12}{M}_2^T)+P_2(W_3-R_{K_1}{W}_3{K}_2{K}_2^{†})M_2^T,\hfill \\ \hfill X_1=& A_0^{†}J-A_0^{†}{A}_1{Q}_2(K_1^{†}{P}_2^{T}J{N}_2-K_1^{†}{W}_3{K}_2+L_{K_1}{W}_4)N_2^T-A_0^{†}{P}_1(V_{11}{M}_1^T+V_{12}{M}_2^T)B_1\hfill \\ \hfill & -A_0^{†}{R}_1{B}_0+L_{A_0}{R}_2+A_0^{†}{A}_1(A_0^{†}{P}_1{V}_{11}{M}_1^T{B}_0-Q_2{V}_{23}{N}_1^T),\hfill \\ \hfill Y_1=& R_{A_0}J{B}_0^{†}+R_{A_0}{A}_1\left(A_0^{†}{P}_1{V}_{11}{M}_1^T{B}_0-Q_2{V}_{23}{N}_1^T\right)B_0^{†}-P_2{R}_{K_1}{P}_2^{T}J{N}_2{K}_2^{†}{M}_2^T{B}_1{B}_0^{†}\hfill \\ \hfill & -P_2\left(W_3-R_{K_1}{W}_3{K}_2{K}_2^{†}\right)M_2^T{B}_1{B}_0^{†}+R_1-R_{A_0}{R}_1{B}_0{B}_0^{†},\hfill \end{array}$$

where $V_{11}$, $V_{12}$, $V_{23}$, $W_3$, $W_4$, $R_1$, and $R_2$ are arbitrary matrices over $\mathbb{D}$ with appropriate sizes.

At the same time, due to the excellent property that dual quaternions can represent both rotation and translation, the theory of dual quaternions is not only one of the most powerful tools for handling rigid-body motion but also finds applications in computer graphics, medical procedures, neural networks, proximity operations in spacecraft, modern robotics, and so on (see [156,157,158]).

Definition 13 

([148]). A dual quaternion is defined as

$$\widehat{q}=q_0+q_1\epsilon ,$$

where $q_0,q_1\in \mathbb{H}$. The set of all dual quaternions is denoted by $\mathbb{DH}$, which a noncommutative ring with zero divisors.

Recently, Xie et al. [159], inspired by the hand–eye calibration problem in robotics research, studied Equation (1) over $\mathbb{DH}$.

Theorem 43 

(Theorem 3.1, [159]). Let

$$A=A_0+A_1\epsilon \in {\mathbb{DH}}^{m\times r},B=B_0+B_1\epsilon \in {\mathbb{DH}}^{s\times n},C=C_0+C_1\epsilon \in {\mathbb{DH}}^{m\times n},$$

where $A_i\in {\mathbb{R}}^{m\times r}$, $B_i\in {\mathbb{R}}^{s\times n}$, and $C_i\in {\mathbb{R}}^{m\times n}$ ($i=0,1$). Set

$$\begin{array}{cc}\hfill & A_{11}=A_1{L}_{A_0},A_2=R_{A_0}{A}_{11},A_3=R_{A_0},C_3=-R_{A_0}{C}_0,B_2=R_{B_0}{B}_1,\hfill \\ \hfill & A_4=R_{A_2}{R}_{A_0},A_5=-A_4{A}_1{A}_0^{†},C_4=A_4(A_1{A}_0^{†}{C}_0+C_0{B}_0^{†}{B}_1-C_1),\hfill \\ \hfill & A_6=R_{A_4}{A}_5,C_5=R_{A_4}{C}_4,B_3=B_2{L}_{B_0},B_4=C_4{L}_{B_0}.\hfill \end{array}$$

Then, the following are equivalent:

(1) 

Equation (1) has a solution pair $X\in {\mathbb{DH}}^{r\times n}$ and $Y\in {\mathbb{DH}}^{m\times s}$;

(2) 

$C_3{L}_{B_0}=0$ and $B_4{L}_{B_3}=0$;

(3) 

The following rank equations hold:

$$\begin{array}{cc}\hfill & \mathrm{rank}\left[\begin{array}{cc}{B}_0& 0\\ -C_0& A_0\end{array}\right]=\mathrm{rank}\left(B_0\right)+\mathrm{rank}\left(A_0\right),\hfill \\ \hfill & \mathrm{rank}\left[\begin{array}{cccc}{B}_0& 0& 0& 0\\ B_1& B_0& 0& 0\\ -C_1& -C_0& A_0& A_1\\ 0& 0& 0& A_0\end{array}\right]=\mathrm{rank}\left[\begin{array}{cc}{B}_0& 0\\ B_1& B_0\end{array}\right]+\mathrm{rank}\left[\begin{array}{cc}{A}_0& A_1\\ 0& A_0\end{array}\right];\hfill \end{array}$$

in which case,

$$X=X_0+X_1\epsilon \mathit{and}Y=Y_0+Y_1\epsilon$$

with

$$\begin{array}{cc}\hfill X_0& =A_0^{†}(C_0+Y_0{B}_0)+L_{A_0}W,\hfill \end{array}$$

(61)

$$\begin{array}{cc}\hfill X_1& =A_0^{†}[C_1+Y_0{B}_1+Y_1{B}_0-A_1{A}_0^{†}(C_0+Y_0{B}_0)-A_{11}W]+L_{A_0}{W}_1,\hfill \\ \hfill Y_0& =A_3^{†}{C}_3{B}_0^{†}+L_{A_3}{U}_1+U_2{R}_{B_0},\hfill \\ \hfill Y_1& =A_4^{†}(C_4-A_5{U}_1{B}_0-A_4{U}_2{B}_2)B_0^{†}+L_{A_4}{W}_3+W_4{R}_{B_0},\hfill \\ \hfill W& =A_2^{†}{A}_3[C_1+Y_0{B}_1+Y_1{B}_0-A_1{A}_0^{†}(C_0+Y_0{B}_0)]+L_{A_2}{W}_2,\hfill \\ \hfill U_1& =A_6^{†}{C}_5{B}_0^{†}+L_{A_6}{W}_5+W_6{R}_{B_0},U_2=A_4^{†}{B}_4{B}_3^{†}+L_{A_4}{W}_7+W_8{R}_{B_3},\hfill \end{array}$$

(62)

where $W_1,W_2,\dots ,W_8$ are arbitrary matrices over $\mathbb{DH}$ with appropriate sizes.

Remark 42. 

Subsequently, Xie and Wang [160] further investigated a more general form of Equation (1) over $\mathbb{DH}$, namely

$$AX+EXF=CY+D,$$

(63)

where X and Y are unknown. Additionally, the systematic introduction to Equation (63) and its more general forms can be found in the book [161].

Remark 43. 

After the Hamilton quaternions, different concepts of quaternions were proposed, greatly enriching the quaternion theory, such as biquaternion (also called complexified quaternions), split quaternions, commutative quaternions (also called Segre biquaternions or reduced biquaternions), generalized commutative quaternions, degenerate quaternions, degenerate pseudo-quaternions, doubly degenerate quaternions, and quaternion algebras over a field (see [43,162,163,164,165,166,167]).

Interestingly, in Remark 8.2.7 of [168], Pottmann and Wallner introduced the concept of the generalized quaternions, which, in specific cases, coincide with Hamilton quaternions, split quaternions, degenerate quaternions, pseudo-degenerate quaternions, and doubly degenerate quaternions.

The theory of dual numbers has also developed rapidly, giving rise to three-dimensional dual numbers (also known as hyper-dual numbers), n-dimensional dual numbers (also known as higher dimensional dual numbers), interval dual numbers, fuzzy dual numbers, neutrosophic dual numbers, and finite complex modulo integer neutrosophic dual numbers (see [169] and references therein).

One can see that dual quaternions are essentially a combination of dual numbers and quaternions. It is then natural to ask: given such a rich variety of quaternions and dual numbers, what sparks will fly when they interact, and what applications will emerge?

6.4. Linear Operator Equations on Hilbert Spaces

Generalizing matrix equations to operator equations on Hilbert spaces or Hilbert $C^{*}$-modules has been a mainstream research direction. For a $C^{*}$-algebra $\mathcal{A}$, a Hilbert $C^{*}$-module $\mathcal{E}$ [170,171] is a right $\mathcal{A}$-module equipped with an $\mathcal{A}$-valued inner product

$$\langle ·,·\rangle :\mathcal{E}\times \mathcal{E}\to \mathcal{A}$$

such that its induced norm $\parallel x\parallel ={\parallel \langle x,x\rangle \parallel }^{{\displaystyle \frac{1}{2}}}$ is complete.

The theory of generalized inverses also serves as an effective tool for studying operator equations. Notably, most research requires the condition that the ranges of related operators are closed to ensure the existence of their MP inverses (see [172,173]). Interestingly, Douglas [174] pioneered an alternative approach in Hilbert spaces without the strong condition of closed ranges, which is known as the Douglas theorem. This work has provided valuable inspiration for subsequent research on operator equations on Hilbert $C^{*}$-modules (see [171,175]).

Let $\mathcal{A}$ be a $C^{*}$-algebra, and let $\mathcal{E}$ and $\mathcal{F}$ be Hilbert $C^{*}$-modules. The set of all bounded $\mathcal{A}$-linear maps

$$A:\mathcal{E}\to \mathcal{F}$$

is denoted by $\mathcal{L}(\mathcal{E},\mathcal{F})$. Particularly, $\mathcal{L}\left(\mathcal{E}\right)=\mathcal{L}(\mathcal{E},\mathcal{E})$. The adjoint of $A\in \mathcal{L}(\mathcal{E},\mathcal{F})$ is a map $A^{*}\in \mathcal{L}(\mathcal{F},\mathcal{E})$ such that

$$\langle Ax,y\rangle =\langle x,A^{*}y\rangle \mathrm{for}\mathrm{all}x\in \mathcal{E},y\in \mathcal{F}.$$

The range and the null space of an operator A are denoted by $\mathcal{R}\left(A\right)$ and $\mathcal{N}\left(A\right)$, respectively. A closed submodule $\mathcal{L}$ of $\mathcal{E}$ is called to be orthogonally complemented [170] if

$$\mathcal{E}=\mathcal{L}\oplus {\mathcal{L}}^{\perp },$$

where ${\mathcal{L}}^{\perp }=\{x\in \mathcal{E}:\langle x,y\rangle =0\mathrm{for}\mathrm{all}y\in \mathcal{L}\}$. And, $\overline{\mathcal{L}}$ is the closure of $\mathcal{L}$. Let $P_{A^{*}}$ be the projection of $\mathcal{E}$ onto $\overline{\mathcal{R}\left(A^{*}\right)}$. And, $R_A=I-P_{A^{*}}$, where I is the identity operator on $\mathcal{E}$.

Let $A,B,C\in \mathcal{L}\left(\mathcal{E}\right)$ be such that A and B are adjointable. Mousavi et al. [175] investigated Equation (5) in Hilbert $C^{*}$-modules $\mathcal{E}$, where only the range closures of adjointable operators need to be orthogonally complemented.

Theorem 44 

(Theorem 3.3, [175]). Let $\overline{\mathcal{R}\left(A\right)}$, $\overline{\mathcal{R}\left(B\right)}$, $\overline{\mathcal{R}\left(A^{*}\right)}$, and $\overline{\mathcal{R}\left(B^{*}\right)}$ are orthogonally complemented. If

$$\mathcal{R}\left(C{R}_B\right)\subseteq \mathcal{R}\left(A\right)\mathrm{and}\mathcal{R}\left(P_{B^{*}}{C}^{*}\right)\subseteq \mathcal{R}\left(B^{*}\right),$$

then Equation (5) is consistent, in which case,

$$X=X_h+X_p\mathit{and}Y=Y_h+Y_p,$$

where $X_p$ and $Y_p$ satisfying $A{X}_p=P_{A}C{R}_B$ and $B^{*}{Y}_p^{*}=P_{B^{*}}{C}^{*}$,

$$X_h=R_A{W}_1+W_2{P}_{B^{*}}\mathit{and}{Y}_h=P_A{W}_3+W_4{R}_{B^{*}},$$

where $W_1,W_2,W_3,W_4\in \mathcal{L}\left(\mathcal{E}\right)$ are arbitrary satisfying $A{W}_2{P}_{B^{*}}+P_A{W}_{3}B=0.$

Remark 44. 

In Example 2.1 of [175], Mousavi et al. gave the example to show that, in a Hilbert $C^{*}$-module, an operator’s range closure being orthogonally complemented is weaker than its range being closed.

Let $A\in \mathcal{L}(\mathcal{E},\mathcal{F})$ satisfy that $\mathcal{R}\left(A\right)$ is closed. In view of the orthogonal decompositions of closed submodules, i.e.,

$$\mathcal{E}=\mathcal{R}\left(A^{*}\right)\oplus \mathcal{N}\left(A\right)\mathrm{and}\mathcal{F}=\mathcal{R}\left(A\right)\oplus \mathcal{N}\left(A^{*}\right),$$

Karizaki et al. in Corollary 1.2 of [176] showed that the operator A can be decomposed into the following matrix form

$$A=\left[\begin{array}{cc}{A}_1& 0\\ 0& 0\end{array}\right]:\left[\begin{array}{c}\mathcal{R}\left(A^{*}\right)\\ \mathcal{N}\left(A\right)\end{array}\right]\to \left[\begin{array}{c}\mathcal{R}\left(A\right)\\ \mathcal{N}\left(A^{*}\right)\end{array}\right],$$

where $A_1$ is invertible, and thus the MP inverse $A^{†}\in \mathcal{L}(\mathcal{F},\mathcal{E})$ of A is

$$A^{†}=\left[\begin{array}{cc}{A}_1^{-1}& 0\\ 0& 0\end{array}\right]:\left[\begin{array}{c}\mathcal{R}\left(A\right)\\ \mathcal{N}\left(A^{*}\right)\end{array}\right]\to \left[\begin{array}{c}\mathcal{R}\left(A^{*}\right)\\ \mathcal{N}\left(A\right)\end{array}\right].$$

Interestingly, four years after [175], Moghani et al. [177] supplemented the conclusions on solving operator Equation (5) on Hilbert $C^{*}$-modules using the matrix forms of adjointable operators and generalized inverses.

Theorem 45 

(Theorem 3.2, [177]). Let $A\in \mathcal{L}(\mathcal{E},\mathcal{F})$, $B\in \mathcal{L}(\mathcal{F},\mathcal{E})$, and $C\in \mathcal{L}\left(\mathcal{F}\right)$ be such that $\mathcal{R}\left(A\right)$ and $\mathcal{R}\left(B\right)$ are closed, $\mathcal{R}\left(A\right)=\mathcal{R}\left(B^{*}\right)$, and $\mathcal{R}\left(A^{*}\right)=\mathcal{R}\left(B\right)$. Then, the operator Equation (5) has a solution pair $X\in \mathcal{L}(\mathcal{F},\mathcal{E})$ and $Y\in \mathcal{L}(\mathcal{E},\mathcal{F})$ if and only if

$$(I-A{A}^{†})C(I-B^{†}B)=0,$$

in which case,

$$\begin{array}{cc}\hfill X& ={\displaystyle \frac{1}{2}}{A}^{†}C+{\displaystyle \frac{1}{2}}{A}^{†}C(I-B^{†}B)+{\displaystyle \frac{1}{2}}WB+(I-A^{†}A)Z,\hfill \\ \hfill Y& ={\displaystyle \frac{1}{2}}A{A}^{†}C{B}^{†}+(I-A{A}^{†})C{B}^{†}-{\displaystyle \frac{1}{2}}AWB{B}^{†}+V(I-B{B}^{†}),\hfill \end{array}$$

where $Z\in \mathcal{L}(\mathcal{F},\mathcal{E})$ and $V\in \mathcal{L}(\mathcal{E},\mathcal{F})$ are arbitrary, and $W\in \mathcal{L}\left(\mathcal{E}\right)$ satisfies

$$(I-A{A}^{†})WB{B}^{†}=0.$$

Remark 45. 

In Section 4 of [177], Moghani et al. further studied the following operator equation

$$AXE+FYB=C,$$

where X and Y are unknown operators between Hilbert $C^{*}$-modules.

Let $\mathcal{H}$ be an infinite dimensional separable Hilbert space. Recently, An et al. [178] revisited the operator Equation (1) in $\mathcal{H}$. For $A\in \mathcal{L}\left(\mathcal{H}\right)$, a $(1,2)$-inverse $A^{(1,2)}$ of A is an operator in $\mathcal{L}\left(\mathcal{H}\right)$ satisfying $A{A}^{(1,2)}A=A$ and $A^{(1,2)}A{A}^{(1,2)}=A^{(1,2)}$.

Theorem 46 

(Theorem 2.1, [178]). Let A, B, and $C\in \mathcal{L}\left(\mathcal{H}\right)$. Then, Equation (1) has a solution pair

$$X=A^{(1,2)}C\mathit{and}Y=-(I-A{A}^{(1,2)})C{B}^{(1,2)}$$

if and only if

$$(I-A{A}^{(1,2)})C(I-B^{(1,2)}B)=0.$$

For $A,B\in \mathcal{L}\left(\mathcal{H}\right)$, we call that the operator pair $(A,B)$ has the generalized Fuglede-Putnam property [178,179] if

$$AX=YB\mathrm{for}X,Y\in \mathcal{L}\left(\mathcal{H}\right)\Rightarrow A^{*}X=Y{B}^{*}.$$

An et al. [178] then presented an interesting connection between the generalized orthogonality and the solvability of the operator Equation (1) in $\mathcal{H}$.

Theorem 47 

(Theorem 2.9, [178]). Let A, B, and $C\in \mathcal{L}\left(\mathcal{H}\right)$.

(1) 

If the operator Equation (1) is consistent, then invertible operators $U,V\in \mathcal{H}\oplus \mathcal{H}$ exist such that

$$U\left[\begin{array}{cc}A& 0\\ 0& B\end{array}\right]=\left[\begin{array}{cc}A& C\\ 0& B\end{array}\right]V.$$

(2) 

Suppose that $(B,A)$ and $(B,B)$ satisfy the generalized Fuglede–Putnam property.

If there exist invertible operators $T,S\in \mathcal{H}\oplus \mathcal{H}$ such that

$$T\left[\begin{array}{cc}A& 0\\ 0& B\end{array}\right]=\left[\begin{array}{cc}A& C\\ 0& B\end{array}\right]S,$$

and the $(2,2)$-entry of $S{T}^{*}$ is invertible, then the operator Equation (1) is consistent.

Example 1. 

Note that Olshevsky in Section 2 of [22] designed an example to show that RET does not hold in infinite dimensional spaces. Indeed, let $\mathcal{H}$ be an infinite dimensional separable Hilbert space with the orthonormal basis ${\left\{e_i\right\}}_{i=1}^{\infty }$. Define the operators $A,C\in \mathcal{L}\left(\mathcal{H}\right)$ as

$$\begin{array}{cc}\hfill & A{e}_{3k+1}=0,A{e}_{3k+2}=0,A{e}_{3k+3}=e_{3k+2}(k=0,1,2,\dots ),\hfill \\ \hfill & C{e}_1=e_1,\mathit{and}C{e}_i=0(i\ne 1).\hfill \end{array}$$

Put $B=A$. Let

$$E=\left[\begin{array}{cc}A& 0\\ 0& B\end{array}\right]\mathit{and}F=\left[\begin{array}{cc}A& C\\ 0& B\end{array}\right].$$

Then, one can observe that the operator E has the only the eigenvalue ${\lambda }_0=0$. Corresponding to this eigenvalue, there are a countable number of Jordan chains of lengths 1 and 2. Additionally, the vectors of these chains form the orthonormal basis of $\mathcal{H}\oplus \mathcal{H}$. So, F has the same properties. Thus, E and F are equivalent. On the other hand, the operator Equation (1) is not solvable. In fact, assume that X and Y satisfy Equation (1). Then, $AX{e}_1=e_1$, which is a contradiction with $e_1\notin \mathcal{R}\left(A\right)$.

Inspired by Bhatia’s characterizations of the unique solution of Sylvester equation (i.e., Theorem VII.2.3, [180]), An et al. [178] further proposed an integral expression for the solution of the operator Equation (1) under the specific conditions. For $A\in \mathcal{L}\left(\mathcal{H}\right)$, $\sigma \left(A\right)$ denotes the spectrum of A.

Theorem 48 

(Theorem 2.17, [178]). Let A, B, and $C\in \mathcal{L}\left(\mathcal{H}\right)$.

(1) 

If the spectra of A and B are contained in the open right half-plane and the open left half-plane, respectively, then the operator Equation (1) has the solution pair

$$X={\int }_0^{\infty }{e}^{-2tA}C\mathrm{d}t\mathit{and}Y=-{\int }_0^{\infty }C{e}^{-2tB}\mathrm{d}t.$$

(2) 

Suppose that A and B are Hermitian operators such that

$$\sigma \left(A\right)\cap \sigma \left(B\right)=\varnothing \mathit{and}\alpha +{\displaystyle \frac{1}{2}}=\beta ,$$

where α and β are eigenvalues of A and B, respectively. Assume that, for an absolutely integrable function f defined on $\mathbb{R}$, its Fourier transform $\widehat{f}\left(s\right)$ satisfies

$$\widehat{f}\left(s\right)={\displaystyle \frac{1}{s}},$$

where $s\in \sigma \left(A\right)-\sigma \left(B\right)$. Then, the operator Equation (1) has the solution pair

$$X={\int }_{-\infty }^{\infty }{e}^{-itA}Cf\left(t\right)\mathrm{d}t\mathit{and}Y={\int }_{-\infty }^{\infty }C{e}^{itB}f\left(t\right)\mathrm{d}t.$$

6.5. Tensor Equations

From the end of the 19th century to the present, the understanding of tensors in multilinear algebra and physics has essentially gone through three ways: as multi-indexed objects satisfying certain transformation rules, as multilinear maps, and as elements in the tensor product of vector spaces (see [181] and references therein). On the other hand, the direct interpretation of “tensors” as multidimensional arrays (or hypermatrices) has also been widely accepted by many scholars (see [182,183,184] and references therein). Following this habit, the tensors in this paper are referred to as multidimensional arrays.

Definition 14. 

Let $\mathbb{F}$ be a ring. An N-order $I_1\times \dots \times I_N$-dimension tensor $\mathcal{A}$ over $\mathbb{F}$ is defined as a multidimensional array with $I_1{I}_2\dots I_N$ entries, i.e.,

$$\mathcal{A}={\left[\begin{array}{c}{a}_{i_1\dots i_N}\end{array}\right]}_{1\le i_j\le I_j}(j=1,\dots ,N),$$

where $a_{i_1\dots i_N}\in \mathbb{F}$ for $1\le i_j\le I_j$ and $j=1,\dots ,N$. Moreover, denote

$${\left(\mathcal{A}\right)}_{i_1\dots i_N}=a_{i_1\dots i_N}.$$

The set of all N-order $I_1\times \dots \times I_N$-dimensional tensors over $\mathbb{F}$ is denoted by ${\mathbb{F}}^{I_1\times \dots \times I_N}$.

Tensor theory has been effectively applied in diverse fields: image processing [185], handwritten digit classification [186], hypergraphs [182], extreme learning machines [187], signal processing and machine learning [188], quantum physics and mechanics [189], etc. In addition, the review article [190] by Kolda and Bader introduced the theory of tensor decomposition and its applications in psychometrics, chemometrics, numerical linear algebra, computer vision, numerical analysis, neuroscience, and so on.

In the 2017 preprint [191], He et al. first discussed SVD and the MP inverse of quaternion tensors under the Einstein product, establishing a fundamental framework for solving quaternion tensor equations under the Einstein product.

Definition 15 

([192]). Let

$$\mathcal{A}=\left[\begin{array}{c}{a}_{i_1\dots i_N{j}_1\dots j_N}\end{array}\right]\in {\mathbb{H}}^{I_1\times \dots \times I_N\times J_1\times \dots \times J_N}\mathit{and}\mathcal{B}=\left[\begin{array}{c}{b}_{j_1\dots j_N{k}_1\dots k_M}\end{array}\right]\in {\mathbb{H}}^{J_1\times \dots \times J_N\times K_1\times \dots \times K_M}.$$

The Einstein product of $\mathcal{A}$ and $\mathcal{B}$ is defined as

$$\mathcal{A}{*}_N\mathcal{B}=\left[\begin{array}{c}{c}_{i_1\dots i_N{k}_1\dots k_M}\end{array}\right]\in {\mathbb{H}}^{I_1\times \dots \times I_N\times K_1\times \dots \times K_M},$$

where

$$c_{i_1\dots i_N{k}_1\dots k_M}=\sum _{j_1\dots j_N}{a}_{i_1\dots i_N{j}_1\dots j_N}{b}_{j_1\dots j_N{k}_1\dots k_M},$$

for $1\le i_j\le I_j$, $j=1,\dots ,N$, $1\le k_t\le K_t$, and $t=1,\dots ,M$.

The conjugate transpose ${\mathcal{A}}^{*}$ of $\mathcal{A}=\left[\begin{array}{c}{a}_{i_1\dots i_N{j}_1\dots j_M}\end{array}\right]\in {\mathbb{H}}^{I_1\times \dots \times I_N\times J_1\times \dots \times J_M}$ is

$${\mathcal{A}}^{*}=\left[\begin{array}{c}{b}_{j_1\dots j_M{i}_1\dots i_N}\end{array}\right]\in {\mathbb{H}}^{J_1\times \dots \times J_M\times I_1\times \dots \times I_N}\mathrm{with}{b}_{j_1\dots j_M{i}_1\dots i_N}={\overline{a}}_{i_1\dots i_N{j}_1\dots j_M}.$$

The MP inverse [193] of $\mathcal{A}\in {\mathbb{H}}^{I_1\times \dots \times I_N\times J_1\times \dots \times J_N}$ is ${\mathcal{A}}^{†}\in {\mathbb{H}}^{J_1\times \dots \times J_N\times I_1\times \dots \times I_N}$ satisfying

$$\mathcal{A}{*}_N{\mathcal{A}}^{†}{*}_N\mathcal{A}=\mathcal{A},{\mathcal{A}}^{†}{*}_N\mathcal{A}{*}_N{\mathcal{A}}^{†}={\mathcal{A}}^{†},{\left(\mathcal{A}{*}_N{\mathcal{A}}^{†}\right)}^{*}=\mathcal{A}{*}_N{\mathcal{A}}^{†},{\left({\mathcal{A}}^{†}{*}_N\mathcal{A}\right)}^{*}={\mathcal{A}}^{†}{*}_N\mathcal{A}.$$

Moreover, let the unit (or identity) tensor be

$${\mathcal{I}}_{I_N}=\left[\begin{array}{c}{e}_{i_1\dots i_N{j}_1\dots j_N}\end{array}\right]\in {\mathbb{H}}^{I_1\times \dots \times I_N\times I_1\times \dots \times I_N},$$

where all diagonal entries $e_{i_1\dots i_N{i}_1\dots i_N}$ are 1 and all off-diagonal entries are 0. Definite

$${\mathcal{L}}_{\mathcal{A}}=\mathcal{I}-{\mathcal{A}}^{†}{*}_N\mathcal{A}\mathrm{and}{\mathcal{R}}_{\mathcal{A}}=\mathcal{I}-\mathcal{A}{*}_N{\mathcal{A}}^{†},$$

where $\mathcal{I}$ denotes the unit tensor with appropriate dimensions.

Theorem 49 

(Corollary 5.3, [191]). Let

$$\mathcal{A}\in {\mathbb{H}}^{I_1\times \dots \times I_N\times J_1\times \dots \times J_N},\mathcal{B}\in {\mathbb{H}}^{H_1\times \dots \times H_M\times L_1\times \dots \times L_M},\mathcal{C}\in {\mathbb{H}}^{I_1\times \dots \times I_N\times L_1\times \dots \times L_M}.$$

Then, Equation (5) over quaternion tensors under the Einstein product, i.e.,

$$\mathcal{A}{*}_N\mathcal{X}+\mathcal{Y}{*}_M\mathcal{B}=\mathcal{C}$$

(64)

has a solution pair

$$\mathcal{X}\in {\mathbb{H}}^{J_1\times \dots \times J_N\times L_1\times \dots \times L_M}\mathit{and}\mathcal{Y}\in {\mathbb{H}}^{I_1\times \dots \times I_N\times H_1\times \dots \times H_M}$$

if and only if

$${\mathcal{R}}_{\mathcal{A}}{*}_N\mathcal{C}{*}_M{\mathcal{L}}_{\mathcal{B}}=0,$$

in which case,

$$\begin{array}{cc}\hfill \mathcal{X}& ={\mathcal{A}}^{†}{*}_N\mathcal{C}-{\mathcal{U}}_1{*}_M\mathcal{B}+{\mathcal{L}}_{\mathcal{A}}{*}_N{\mathcal{U}}_2,\hfill \\ \hfill \mathcal{Y}& ={\mathcal{R}}_{\mathcal{A}}{*}_N\mathcal{C}{*}_M{\mathcal{B}}^{†}+\mathcal{A}{*}_N{\mathcal{U}}_1+{\mathcal{U}}_3{*}_M{\mathcal{R}}_{\mathcal{B}},\hfill \end{array}$$

where ${\mathcal{U}}_1$, ${\mathcal{U}}_2$, and ${\mathcal{U}}_3$ are arbitrary tensors over $\mathbb{H}$ with appropriate dimensions.

Remark 46. 

(1) 

Theorem 49 is a direct corollary of Theorem 5.1 of [191], which establishes the solvability conditions and the general solution for the following quaternion tensor equation:

$$\mathcal{A}{*}_N\mathcal{X}{*}_M\mathcal{D}+\mathcal{E}{*}_N\mathcal{Y}{*}_M\mathcal{B}=\mathcal{C},$$

where $\mathcal{X}$ and $\mathcal{Y}$ are unknown and other tensors are given over $\mathbb{H}$.

(2) 

Inspired by the transformation between tensors and matrices over $\mathbb{R}$ (see (Definition 2.8, [194])), He et al. [191,195] defined an analogous transformation over $\mathbb{H}$, i.e., the transformation f is a map defined as

$$\begin{array}{cc}\hfill f:& {\mathbb{H}}^{I_1\times \dots \times I_N\times J_1\times \dots \times J_N}\to {\mathbb{H}}^{(I_1{I}_2\dots ·I_N)\times (J_1{J}_2\dots ·J_N)}\hfill \\ \hfill & \mathcal{A}\in {\mathbb{H}}^{I_1\times \dots \times I_N\times J_1\times \dots \times J_N}\mapsto A=f\left(\mathcal{A}\right)\in {\mathbb{H}}^{(I_1{I}_2\dots I_N)\times (J_1{J}_2\dots J_N)},\hfill \end{array}$$

where the components of A are given by

$${\left(\mathcal{A}\right)}_{i_1\dots i_N{j}_1\dots j_N}\stackrel{f}{\mapsto }{\left(A\right)}_{\left[i_1+{\sum }_{k=2}^N(i_k-1){\prod }_{s=1}^{k-1}{I}_s\right]\left[j_1+{\sum }_{k=2}^N(j_k-1){\prod }_{s=1}^{k-1}{J}_s\right]}.$$

Lemma 2.2 of [191] shows that the transformation f is a bijection satisfying

$$f(\mathcal{A}+\mathcal{B})=f\left(\mathcal{A}\right)+f\left(\mathcal{B}\right)\mathit{and}f\left(\mathcal{A}{*}_N\mathcal{C}\right)=f\left(\mathcal{A}\right)f\left(\mathcal{C}\right)$$

for $\mathcal{A},\mathcal{B}\in {\mathbb{H}}^{I_1\times \dots \times I_N\times J_1\times \dots \times J_N}$ and $\mathcal{C}\in {\mathbb{H}}^{J_1\times \dots \times J_N\times L_1\times \dots \times L_N}$. The transformation f ingeniously bridges quaternion tensors under the Einstein product and quaternion matrices under the ordinary product. By virtue of its isomorphism property, f serves as a powerful tool for studying problems related to quaternion tensors under the Einstein product.

(3) 

The work [191] on quaternion tensor equations has profoundly influenced subsequent research on tensor equations over $\mathbb{H}$ (see [196,197,198,199,200,201,202]).

Subsequently, Wang et al. [125] discussed the minimum-norm least-squares solution of the quaternion tensor equation of the form (64), i.e.,

$$\mathcal{A}{*}_N\mathcal{X}+\mathcal{Y}{*}_N\mathcal{B}=\mathcal{D}$$

with the unknown $\mathcal{X}$ and $\mathcal{Y}$. We now introduce some notations.

Let $\mathcal{A}=\left[a_{i_1\dots i_N{j}_1\dots j_M}\right]\in {\mathbb{H}}^{I_1\times \dots \times I_N\times J_1\times \dots \times J_M}$. The transpose ${\mathcal{A}}^T$ of $\mathcal{A}$ is

$${\mathcal{A}}^T=\left[b_{j_1\dots j_M{i}_1\dots i_N}\right]\in {\mathbb{H}}^{J_1\times \dots \times J_M\times I_1\times \dots \times I_N},$$

where $b_{j_1\dots j_M{i}_1\dots i_N}=a_{i_1\dots i_N{j}_1\dots j_M}$, and the conjugate $\overline{\mathcal{A}}$ of $\mathcal{A}$ is

$$\overline{\mathcal{A}}=\left[{\overline{a}}_{i_1\dots i_N{j}_1\dots j_M}\right]\in {\mathbb{H}}^{I_1\times \dots \times I_N\times J_1\times \dots \times J_M}.$$

The symbol

$${\mathcal{A}}_{(i_1\dots i_N|:)}=\left[a_{i_1\dots i_N:\dots :}\right]\in {\mathbb{H}}^{J_1\times \dots \times J_M}$$

stands for a subblock of $\mathcal{A}$, and $\mathrm{Vec}\left(\mathcal{A}\right)$ is a new tensor obtained by lining up all subtensor in a column, where the t-th subblock of $\mathrm{Vec}\left(\mathcal{A}\right)$ is ${\mathcal{A}}_{(i_1\dots i_N|:)}$ for

$$t=i_N+\sum _{K=1}^{N-1}\left[(i_K-1)\prod _{L=K+1}^N{I}_L\right].$$

Since $\mathcal{A}={\mathcal{A}}_1+{\mathcal{A}}_2\mathbf{j}$ for ${\mathcal{A}}_1,{\mathcal{A}}_2\in {\mathbb{C}}^{I_1\times \dots \times I_N\times J_1\times \dots \times J_M}$, the complex representation tensor of $\mathcal{A}$ is defined as

$$f\left(\mathcal{A}\right)=\left[\begin{array}{cc}{\mathcal{A}}_1& {\mathcal{A}}_2\\ -{\overline{\mathcal{A}}}_2& {\overline{\mathcal{A}}}_1\end{array}\right]\in {\mathbb{C}}^{2I_1\times \dots \times 2I_N\times 2J_1\times \dots \times 2J_M}.$$

Let ${\mathsf{\Theta }}_{\mathcal{A}}=[{\mathcal{A}}_1,{\mathcal{A}}_2]$, $\overrightarrow{\mathcal{A}}=(\mathrm{Re}{\mathcal{A}}_1,\mathrm{Im}{\mathcal{A}}_1,\mathrm{Re}{\mathcal{A}}_2,\mathrm{Im}{\mathcal{A}}_2)$,

$$\mathrm{Vec}\left(\overrightarrow{\mathcal{A}}\right)=\left[\begin{array}{c}\mathrm{Vec}\left(\mathrm{Re}{\mathcal{A}}_1\right)\\ \mathrm{Vec}\left(\mathrm{Im}{\mathcal{A}}_1\right)\\ \mathrm{Vec}\left(\mathrm{Re}{\mathcal{A}}_2\right)\\ \mathrm{Vec}\left(\mathrm{Im}{\mathcal{A}}_2\right)\end{array}\right],\mathrm{and}{\mathcal{K}}_{J_M}=\left[\begin{array}{cccc}\mathcal{I}& \mathbf{i}{\mathcal{I}}_{J_M}& 0& 0\\ 0& 0& {\mathcal{I}}_{J_M}& \mathbf{i}{\mathcal{I}}_{J_M}\\ {\mathcal{I}}_{J_M}& -\mathbf{i}{\mathcal{I}}_{J_M}& 0& 0\\ 0& 0& {\mathcal{I}}_{J_M}& -\mathbf{i}{\mathcal{I}}_{J_M}\end{array}\right],$$

where $\mathbf{i}$ is imaginary unit such that ${\mathbf{i}}^2=-1$.

Let $\mathcal{A}\in {\mathbb{R}}^{I_1\times \dots \times I_N\times J_1\times \dots \times J_N}$. The Frobenius norm ${\parallel ·\parallel }_F$ of $\mathcal{A}$ is

$${\parallel \mathcal{A}\parallel }_F={\left(\sum _{i_1\dots i_N{j}_1\dots j_N}{\left|a_{i_1\dots i_N{j}_1\dots j_N}\right|}^2\right)}^{1/2}.$$

For $\mathcal{B}=B_1+B_2\mathrm{j}\in {\mathbb{H}}^{J_1\times \dots \times J_N\times K_1\times \dots \times K_M}$, define

$$\mathcal{A}•f\left(\mathcal{B}\right)=\left[\begin{array}{cc}\mathcal{A}\otimes B_1& \mathcal{A}\otimes B_2\\ \mathcal{A}\otimes (-{\overline{B}}_2)& \mathcal{A}\otimes {\overline{B}}_1\end{array}\right],$$

where ⊗ denotes the Kronecker product of two tensors. The the inverse of $\mathcal{A}\in {\mathbb{C}}^{I_1\times \dots \times I_N\times I_1\times \dots \times I_N}$ is the tensor ${\mathcal{A}}^{-1}\in {\mathbb{C}}^{I_1\times \dots \times I_N\times I_1\times \dots \times I_N}$ satisfying

$$\mathcal{A}{*}_N{\mathcal{A}}^{-1}={\mathcal{A}}^{-1}{*}_N\mathcal{A}=\mathcal{I}.$$

Theorem 50 

(Corollary 3.4, [125]). Let $\mathcal{A},\mathcal{B},\mathcal{D}\in {\mathbb{H}}^{J_1\times \dots \times J_N\times J_1\times \dots \times J_N}$, and let

$$\begin{array}{c}\hfill {\mathbb{H}}_{L1}=\left\{\left[\mathcal{X},\mathcal{Y}\right]\right\}\end{array}$$

be the set of all $\mathcal{X},\mathcal{Y}\in {\mathbb{H}}^{J_1\times \dots \times J_N\times J_1\times \dots \times J_N}$ such that

$$\parallel \mathcal{A}{*}_N\mathcal{X}+\mathcal{Y}{*}_N{\mathcal{B}-\mathcal{D}\parallel }_F^2=\underset{{\mathcal{X}}_1,{\mathcal{Y}}_1\in {\mathbb{H}}^{I_1\times \dots \times I_N\times J_1\times \dots \times J_N}}{min}{\parallel \mathcal{A}{*}_N{\mathcal{X}}_1+{\mathcal{Y}}_1{*}_N\mathcal{B}-\mathcal{D}\parallel }_F^2.$$

Denote $\mathcal{A}={\mathcal{A}}_1+{\mathcal{A}}_2\mathbf{j}$. Put

$$\begin{array}{cc}\hfill {\mathcal{P}}_{01}& =\left[\begin{array}{cc}{\mathcal{A}}_1•f{\left({\mathcal{I}}_{J_N}\right)}^T& {\mathcal{A}}_2•f{\left({\mathcal{I}}_{J_N}\mathrm{j}\right)}^{*}\end{array}\right]{*}_N{\mathcal{K}}_{J_N},{\mathcal{Q}}_{01}=\left[\begin{array}{cc}{\mathcal{I}}_{J_N}•f{\left(\mathcal{B}\right)}^T& 0\end{array}\right]{*}_N{\mathcal{K}}_{J_N},\hfill \\ \hfill {\mathcal{T}}_{11}& =\left[\begin{array}{cc}\mathrm{Re}{\mathcal{P}}_{01}& \mathrm{Re}{\mathcal{Q}}_{01}\end{array}\right],{\mathcal{T}}_{12}=\left[\begin{array}{cc}\mathrm{Im}{\mathcal{P}}_{01}& \mathrm{Im}{\mathcal{Q}}_{01}\end{array}\right],{\mathcal{E}}_1=\left[\begin{array}{c}\mathrm{Vec}\left(\mathrm{Re}{\mathsf{\Theta }}_{\mathcal{D}}\right)\\ \mathrm{Vec}\left(\mathrm{Im}{\mathsf{\Theta }}_{\mathcal{D}}\right)\end{array}\right],\hfill \\ \hfill {\mathcal{R}}_1& =\left(\mathcal{I}-{\mathcal{T}}_{11}^{†}{*}_N{\mathcal{T}}_{11}\right){*}_N{\mathcal{T}}_{12}^T,\hfill \\ \hfill {\mathcal{H}}_1& ={\mathcal{R}}_1^{†}+\left(\mathcal{I}-{\mathcal{R}}_1^{†}{*}_N{\mathcal{R}}_1\right){*}_N{\mathcal{Z}}_1{*}_N{\mathcal{T}}_{12}{*}_N{\mathcal{T}}_{11}^{†}{*}_N{\left({\mathcal{T}}_{11}^{†}\right)}^T{*}_N\left(\mathcal{I}-{\mathcal{T}}_{12}^T{*}_N{\mathcal{R}}_1^{†}\right),\hfill \\ \hfill {\mathcal{Z}}_1& ={\left(\mathcal{I}+\left(\mathcal{I}-{\mathcal{R}}_1^{†}{*}_N{\mathcal{R}}_1\right){*}_N{\mathcal{T}}_{12}{*}_N{\mathcal{T}}_{11}^{†}{*}_N{\left({\mathcal{T}}_{11}^{†}\right)}^T{*}_N{\mathcal{T}}_{12}^T{*}_N\left(\mathcal{I}-{\mathcal{R}}_1^{†}{*}_N{\mathcal{R}}_1\right)\right)}^{-1}.\hfill \end{array}$$

(1) 

Then,

$$\begin{array}{cc}\hfill {\mathbb{H}}_{L1}=\left\{[\mathcal{X},\mathcal{Y}]\right|\left[\begin{array}{c}\mathrm{Vec}\left(\overrightarrow{\mathcal{X}}\right)\\ \mathrm{Vec}\left(\overrightarrow{\mathcal{Y}}\right)\end{array}\right]=& \left[\begin{array}{cc}{\mathcal{T}}_{11}^{†}-{\mathcal{H}}_1^{\mathrm{T}}{*}_N{\mathcal{T}}_{12}{*}_N{\mathcal{T}}_{11}^{†}& {\mathcal{H}}_1^{\mathrm{T}}\end{array}\right]{*}_N{\mathcal{E}}_1\hfill \\ \hfill & +\left(\mathcal{I}-{\mathcal{T}}_{11}^{†}{*}_N{\mathcal{T}}_{11}-{\mathcal{R}}_1{*}_N{\mathcal{R}}_1^{†}\right){*}_N{\mathcal{W}}_1\},\hfill \end{array}$$

where ${\mathcal{W}}_1$ is arbitrary with appropriate dimensions.

(2) 

If $[{\mathcal{X}}_{l1},{\mathcal{Y}}_{l1}]\in {\mathbb{H}}_{L1}$ satisfies

$${∥[{\mathcal{X}}_{l1},{\mathcal{Y}}_{l1}]∥}_F^2=\underset{[\mathcal{X},\mathcal{Y}]\in {\mathbb{H}}_{L1}}{min}\left({\parallel \mathcal{X}\parallel }_F^2+{\parallel \mathcal{Y}\parallel }_F^2\right),$$

then $[{\mathcal{X}}_{l1},{\mathcal{Y}}_{l1}]\in {\mathbb{H}}_{L1}$ is unique and

$$\left[\begin{array}{c}\mathrm{Vec}\left(\overrightarrow{{\mathcal{X}}_{l1}}\right)\\ \mathrm{Vec}\left(\overrightarrow{{\mathcal{Y}}_{l1}}\right)\end{array}\right]=\left[\begin{array}{cc}{\mathcal{T}}_{11}^{†}-{\mathcal{H}}_1^{\mathrm{T}}{*}_N{\mathcal{T}}_{12}{*}_N{\mathcal{T}}_{11}^{†}& {\mathcal{H}}_1^{\mathrm{T}}\end{array}\right]{*}_N{\mathcal{E}}_1.$$

Remark 47. 

Recently, some scholars have extended quaternion tensor equations under the Einstein product to different categories. For instance, Jia and Wang [203] investigated split quaternion tensor equations, while Yang et al. [204] explored dual split quaternion tensor equations. On the other hand, tensor theory encompasses a variety of product operations, including the Einstein product [192], k-mode Product [182], contracted product [205], T-product [206], Qt-product [207], general product [208], cosine transform product (c-product) [209], and M-product [210,211]. Combined with Remark 43, the investigation of tensor equations over diverse quaternion algebras under various tensor products reveals significant untapped research potential.

6.6. Polynomial Matrix Equations

As described in Section 3, Roth was the first to study the solvability conditions of polynomial matrix Equation (2) via the equivalence of two block polynomial matrices. Since then, the theoretical research and practical applications regarding this polynomial matrix equation have gradually become more extensive and enriched. For instance, it is successfully applied to multivariable linear discrete systems in stochastic control [212], the algebraic regulator problem [213], etc. Next, we mainly discuss different approaches to studying this polynomial matrix.

6.6.1. By the Divisibility of Polynomials

Let F be a field. Cheng and Pearson [214], in their research on the regulator problem with internal stability, provided an equivalent characterization of the solvability of the polynomial matrix equation

$$BX+YD=P$$

(65)

with given polynomial matrix $B\in {\mathbb{F}}^{p\times n}\left[\lambda \right]$, $D\in {\mathbb{F}}^{p\times p}\left[\lambda \right]$, and $P\in {\mathbb{F}}^{p\times p}\left[\lambda \right]$, by the divisibility of a series of polynomials. Assume that $\mathrm{rank}\left(B\right)=r$ and $\mathrm{rank}\left(D\right)=p$. Using Lemma 2 of [214] (i.e., Smith normal form theorem for polynomial matrices), there exist unimodular matrices $M_b,N_b,M_d$, and $N_d$ such that

$$M_{b}B{N}_b=B_d=\left[\begin{array}{cc}{B}_0& 0\\ 0& 0\end{array}\right]\mathrm{and}{M}_{d}D{N}_d=\mathrm{diag}(d_1,d_2,\dots ,d_p),$$

(66)

where $B_0=\mathrm{diag}(b_1,b_2,\dots ,b_r)$, $b_1{b}_2\dots b_p\ne 0$, and $d_1{d}_2\dots d_p\ne 0$. Left-multiplying (65) by $M_b$ and right-multiplying (65) by $N_d$ yield that

$$B_d{X}^{\prime }+Y^{\prime }{D}_d=P^{\prime },$$

(67)

where $X^{\prime }=N_b^{-1}X{N}_d$, $Y^{\prime }=M_{b}Y{M}_d^{-1}$, and $P^{\prime }=M_{b}P{N}_d$. For $a,b\in \mathbb{F}\left[\lambda \right]$, $a|b$ denotes a divides b.

Theorem 51 

(Lemma 6, [214]). Let B, D, and P be given in (65) with $\mathrm{rank}\left(B\right)=r$ and $\mathrm{rank}\left(D\right)=p$. Let $b_1,b_2,\dots ,b_r$ and $d_1,d_2,\dots ,d_p$ be given in (66), and $P^{\prime }=\left[p_{ij}^{\prime }\right]$ be given in (67). Then, Equation (65) has a solution pair $(X,Y)$ if and only if

(1) 

$g_{ij}|p_{ij}^{\prime }$ for $i=1,\dots ,r$ and $j=1,\dots ,p$;

(2) 

$d_j|p_{ij}^{\prime }$ if $p>r$, for $i=r+1,\dots ,p$ and $j=1,\dots ,p$.

where $g_{ij}$ is the monic greatest common divisor of $b_i$ and $d_j$.

Remark 48. 

Notably, in Theorem 2 of [214], Cheng and Pearson equivalently transform the solvability of a restricted regulator problem with internal stability into the solvability of Equation (65) in a special form.

6.6.2. By Skew-Prime Polynomial Matrices

Let F be a field. In 1978, Wolovich [215] proposed a new approach to studying the polynomial matrix equation

$$A\left(\lambda \right)X\left(\lambda \right)+Y\left(\lambda \right)B\left(\lambda \right)=C\left(\lambda \right)$$

(68)

with given $A\left(\lambda \right)\in {\mathbb{F}}^{p\times m}\left[\lambda \right]$, $B\left(\lambda \right)\in {\mathbb{F}}^{q\times t}\left[\lambda \right]$, and $C\left(\lambda \right)\in {\mathbb{F}}^{p\times t}\left[\lambda \right]$, based on skew-prime polynomial matrices.

Definition 16 

([215,216]). Let $A\left(\lambda \right)\in {\mathbb{F}}^{p\times m}\left[\lambda \right]$ and $B\left(\lambda \right)\in {\mathbb{F}}^{q\times p}\left[\lambda \right]$ with $q+m>p$. Assume that there exist $M\left(\lambda \right)\in {\mathbb{F}}^{m\times p}\left[\lambda \right]$ and $N\left(\lambda \right)\in {\mathbb{F}}^{p\times q}\left[\lambda \right]$ such that

$$A\left(\lambda \right)M\left(\lambda \right)+N\left(\lambda \right)B\left(\lambda \right)=I_p.$$

Then, $A\left(\lambda \right)$ and $B\left(\lambda \right)$ are called externally skew prime (or $B\left(\lambda \right)$ and $A\left(\lambda \right)$ are called internally skew prime). Moreover, $A\left(\lambda \right)$ and $N\left(\lambda \right)$ are called relatively left prime while $M\left(\lambda \right)$ and $B\left(\lambda \right)$ are called relatively right prime.

Suppose that $A\left(\lambda \right)$ is nonsingular with $p=m$. Then, $A^{-1}\left(s\right)C\left(\lambda \right)$ can be factored in dual prime form:

$$A^{-1}\left(s\right)C\left(\lambda \right)=\overline{C}\left(s\right){\overline{\mathcal{A}}}^{-1}\left(s\right),$$

(69)

where $\overline{C}\left(s\right)\in {\mathbb{F}}^{p\times t}\left[\lambda \right]$ and $\overline{\mathcal{A}}\left(s\right)\in {\mathbb{F}}^{t\times t}\left[\lambda \right]$ are relatively right prime.

Theorem 52 

(Theorem 3 and Corollary 3, [215]). Let $A\left(\lambda \right)\in {\mathbb{F}}^{p\times p}\left[\lambda \right]$ be nonsingular and let $\overline{\mathcal{A}}\left(s\right)$ be given in (69).

(1) 

If $\overline{\mathcal{A}}\left(s\right)$ and $B\left(\lambda \right)$ are externally skew, then Equation (68) is consistent.

(2) 

Suppose that $A\left(\lambda \right)$ and $C\left(\lambda \right)$ are relatively left prime. Then, Equation (68) is consistent if and only if $\overline{\mathcal{A}}\left(s\right)$ and $B\left(\lambda \right)$ are externally skew prime.

Remark 49. 

Note that Wolovich proved the sufficiency of the item (2) in Theorem 52 via a constructive method, implying a new procedure to find a solution pair of Equation (68). Moreover, when $C\left(\lambda \right)=I$, all solutions of Equation (68) are characterized in (Section 5, [215]), and this characterization is further used to obtain the unique solution of Equation (68).

6.6.3. By the Realization of Matrix Fraction Descriptions

Let $\mathcal{F}$ be a field. For given $Q\in {\mathbb{F}}^{p\times q}\left[\lambda \right]$, $R\in {\mathbb{F}}^{m\times t}\left[\lambda \right]$, and $\Phi \in {\mathbb{F}}^{p\times t}\left[\lambda \right]$, consider the polynomial matrix equation

$$XR+QY=\Phi ,$$

(70)

where $X\in {\mathbb{F}}^{p\times m}\left[\lambda \right]$ and $Y\in {\mathbb{F}}^{q\times t}\left[\lambda \right]$ are unknown. According to the realization of matrix fraction descriptions presented in [217], Emre and Silverman [218] transformed Equation (70) into a set of linear matrix equations when Q is nonsingular.

Let ${\mathbb{F}}^q\left[\lambda \right]$ and ${\mathbb{F}}^q\left(\lambda \right)$ be the sets of all q-tuples of polynomials in $\lambda$ with coefficients in $\mathbb{F}$ and all q-tuples of rational functions in $\lambda$ over $\mathbb{F}$, respectively. Assume that Q is nonsingular with $p=q$. Let

$${\mathbb{F}}_Q=\{x\in {\mathbb{F}}^p\left[\lambda \right]:Q^{-1}x\mathrm{is}\mathrm{strictly}\mathrm{proper}\}.$$

For $X_1\in {\mathbb{F}}^{p\times m}\left[\lambda \right]$ satisfying that $Q^{-1}{X}_1$ is strictly proper, define the following $\mathbb{F}$-linear maps:

$$\begin{array}{cc}\hfill & G:{\mathbb{F}}^m\to {\mathbb{F}}_Q,u\mapsto X_{1}u\mathrm{for}u\in {\mathbb{F}}^m,\hfill \\ \hfill & \pi :{\mathbb{F}}^p\left(\lambda \right)\to {\mathbb{F}}^p\left(\lambda \right),q\mapsto \mathrm{strictly}\mathrm{proper}\mathrm{part}\mathrm{of}q,\hfill \\ \hfill & {\pi }_Q:{\mathbb{F}}^p\left[\lambda \right]\to {\mathbb{F}}^p\left[\lambda \right],x\mapsto Q\pi \left(Q^{-1}x\right),\hfill \\ \hfill & F:{\mathbb{F}}_Q\to {\mathbb{F}}_Q,x\mapsto {\pi }_Q\left(zx\right),\hfill \\ \hfill & H:{\mathbb{F}}_Q\to {\mathbb{F}}^p,x\mapsto {\left(Q^{-1}x\right)}_{-1},\hfill \end{array}$$

where ${\left(Q^{-1}x\right)}_{-1}$ is the coefficient of ${\lambda }^{-1}$ in the formal power series of $Q^{-1}x$ in ${\lambda }^{-1}$. For $Z=Q^{-1}{X}_1$, we call $\mathsf{\Sigma }=(F,G,H)$ the Q-realization of Z [217].

Let $S\in {\mathbb{F}}^{p\times n}\left[\lambda \right]$ be such that its columns are a basis of ${\mathbb{F}}_Q$. Let $(F,G_1,H)$ be the Q-realization of $Q^{-1}S$. Let $\widehat{F}$, ${\widehat{G}}_1$, and $\widehat{H}$ denote the matrix representations of F, $G_1$, and H, respectively, with respect to the canonical bases of ${\mathbb{F}}^m$ and ${\mathbb{F}}^p$, and the columns of S serving as a basis of ${\mathbb{F}}_Q$. Put

$$R=\sum _{j=0}^r{u}_{-j}{\lambda }^j,$$

where $u_{-j}\in {\mathbb{F}}^{m\times p}$. Define $\widehat{\Phi }\in {\mathbb{F}}^{n\times p}$ uniquely by ${\pi }_Q(\Phi )=S\widehat{\Phi }$, and for the unique polynomial matrix ${\Phi }_1$, express $\Phi$ as

$$\Phi =Q{\Phi }_1+S\widehat{\Phi }.$$

(71)

Moreover, let the linear equations be as follows:

$$\sum _{j=0}^r{\widehat{F}}^j\widehat{G}{u}_{-j}=\widehat{\Phi },$$

(72)

where $\widehat{G}\in {\mathbb{F}}^{n\times m}$ are unknown.

Theorem 53 

(Theorem 2.5, [218]). Let R, Q, and Φ be given in (70). Suppose that Q is nonsingular with $p=q$. Denote

$$\overline{E}(Q,R)=\{(X_1,Y_1)\mid X_{1}R+Q{Y}_1=\Phi \mathit{and}{Q}^{-1}{X}_1\mathit{is}\mathit{strictly}\mathit{proper}\}.$$

The following are equivalent:

(1) 

$(X_1,Y_1)\in \overline{E}(Q,R)$;

(2) 

Equation (72) has a solution $\widehat{G}$ such that

$$X_1=S\widehat{G}\mathit{and}{Y}_1={\Phi }_1-Q_p,$$

where ${\Phi }_1$ satisfies (71), and $Q_p$ is the polynomial part of $Q^{-1}{X}_{1}R$.

Remark 50. 

(1) 

Under the hypotheses of Theorem 53, let

$$E(Q,R)=\left\{\right(X,Y\left)\right|XR+QY=\Phi \}.$$

In terms of Lemma 2.2 of [218], Emre and Silverman have shown that

$$E(Q,R)=\{(X_1,Y_1)+(\overline{X},\overline{Y})\mid (X_1,Y_1)\in \overline{E}(Q,R)\mathit{and}(\overline{X},\overline{Y})\in \overline{H}(Q,R)\},$$

where $\overline{H}(Q,R)=\{(Q{Q}_1,-Q_{1}R)\mid Q_1\mathit{is}\mathit{an}\mathit{arbitrary}\mathit{polynomial}\mathit{matrix}\}$. This implies that, to characterize $E(Q,R)$, it is sufficient to characterize $\overline{E}(Q,R)$.

(2) 

In Section 3, [218], Equation (70) is further generalized to the case where Q is a general polynomial matrix. In fact, for $Q\in {\mathbb{F}}^{p\times q}\left[\lambda \right]$, there exist unimodular polynomial matrices $M_1$ and $M_2$ such that

$$M_{1}Q{M}_2=\left[\begin{array}{cc}\widehat{Q}& 0\\ 0& 0\end{array}\right],$$

where $\widehat{Q}$ is the nonsingular polynomial matrix. Let

$$\widehat{X}=M_{1}X=\left[\begin{array}{c}{\widehat{X}}_1\\ {\widehat{X}}_2\end{array}\right],\widehat{Y}=M_2^{-1}Y=\left[\begin{array}{c}{\widehat{Y}}_1\\ {\widehat{Y}}_2\end{array}\right],M_1\Phi =\left[\begin{array}{c}{\tilde{\Phi }}_1\\ {\tilde{\Phi }}_2\end{array}\right].$$

Then,

$$\begin{array}{cc}\hfill \mathit{Equation}\left(70\right)& \iff \widehat{X}R+\left[\begin{array}{cc}\widehat{Q}& 0\\ 0& 0\end{array}\right]\left[\begin{array}{c}{\widehat{Y}}_1\\ {\widehat{Y}}_2\end{array}\right]=\left[\begin{array}{c}{\tilde{\Phi }}_1\\ {\tilde{\Phi }}_2\end{array}\right]\hfill \\ \hfill & \iff \left\{\begin{array}{c}{\widehat{X}}_{2}R={\tilde{\Phi }}_2,\hfill \\ {\widehat{X}}_{1}R+\widehat{Q}{\widehat{Y}}_1={\tilde{\Phi }}_1.\hfill \end{array}\right.\hfill \end{array}$$

(73)

So, Theorem 53 can be applied to the second equation of (73); see [219] for the first Equation of (73).

6.6.4. By the Unilateral Polynomial Matrix Equation

Let $\mathbb{F}$ be a field. For given $A,B,C\in {\mathbb{F}}^{n\times n}\left[\lambda \right]$, consider the following polynomial matrix Equation (also called the bilateral polynomial matrix equation):

$$AX+YB=C,$$

(74)

where $X,Y\in {\mathbb{F}}^{n\times n}\left[\lambda \right]$ are unknown. Żak [220] proposed an algorithm for finding the unique solution pair $(X,Y)$ of Equation (74). In fact, let

$$A=\sum _{i=0}^N{\lambda }^i{A}_i\mathrm{and}B=\sum _{j=0}^M{\lambda }^j{B}_j.$$

For $C={\left[c_{ij}\right]}_{n\times n}\in {\mathbb{F}}^{n\times n}\left[\lambda \right]$, denote

$${\mathrm{vec}}_{\mathrm{r}}\left(C\right)={\left[\begin{array}{cccccccccc}{c}_{11}& \dots & c_{1n}& c_{21}& \dots & c_{2n}& \dots & c_{n1}& \dots & c_{nn}\end{array}\right]}^T.$$

Using the Kronecker product, Equation (74) can be transformed into the following unilateral polynomial matrix equation:

$$\mathfrak{A}\mathfrak{X}+\mathfrak{B}\mathfrak{Y}=\mathfrak{C},$$

(75)

where $\mathfrak{X}={\mathrm{vec}}_{\mathrm{r}}\left(X\right)$, $\mathfrak{Y}={\mathrm{vec}}_{\mathrm{r}}\left(Y\right)$, $\mathfrak{C}={\mathrm{vec}}_{\mathrm{r}}\left(C\right)$,

$$\begin{array}{cc}\hfill & \mathfrak{A}=\sum _{i=0}^N{\lambda }^i(A_i\otimes I_n),\mathrm{and}\mathfrak{B}=\sum _{j=0}^M{\lambda }^j(I_n\otimes B_j^T).\hfill \end{array}$$

Let

$$X=\sum _{i=0}^{M_1-1}{\lambda }^i{X}_i\mathrm{and}Y=\sum _{i=0}^{N_1-1}{\lambda }^i{Y}_i.$$

Denote

$${\mathfrak{A}}_i=A_i\otimes I_n\mathrm{and}{\mathfrak{B}}_j=I_n\otimes B_j^T,$$

where $i=0,1,\dots ,N$ and $j=0,1,\dots ,M$. Then, by comparing of like powers, (75) can be rewritten as

$$\begin{array}{c}\left[\begin{array}{cccccc}{\mathfrak{B}}_0& & & {\mathfrak{A}}_0& & \\ {\mathfrak{B}}_1& {\mathfrak{B}}_0& & {\mathfrak{A}}_1& {\mathfrak{A}}_0& \\ ⋮& {\mathfrak{B}}_1& {\mathfrak{B}}_0& ⋮& {\mathfrak{A}}_1& {\mathfrak{A}}_0\\ {\mathfrak{B}}_M& ⋮& {\mathfrak{B}}_1& {\mathfrak{A}}_N& ⋮& {\mathfrak{A}}_1\\ & {\mathfrak{B}}_M& ⋮& & {\mathfrak{A}}_N& ⋮\\ & & {\mathfrak{B}}_M& & & {\mathfrak{A}}_N\end{array}\right]\left[\begin{array}{c}{Y}_0\\ ⋮\\ Y_{N_1-1}\\ X_0\\ ⋮\\ X_{M_1-1}\end{array}\right]=\left[\begin{array}{c}{C}_0\\ C_1\\ ⋮\\ ⋮\\ ⋮\end{array}\right].\hfill \\ N_1+M_1\mathrm{blocks}\hfill \end{array}$$

(76)

Let us assume without loss of generality that $B_0$ is nonsingular, which implies that ${\mathfrak{B}}_0$ is also nonsingular. As shown by Feinstein and Bar-Ness in [221], performing a series of elementary row operations on Equation (76) yields the following form:

$$\left[\begin{array}{cc}\begin{array}{ccc}{I}_n& & \\ & \ddots & \\ & & I_n\end{array}& \begin{array}{ccc}{L}_0& & \\ L_1& & \\ ⋮& & \\ L_0& \dots & L_0\end{array}\\ 0& L\end{array}\right]\left[\begin{array}{c}y\\ x\end{array}\right]=\left[\begin{array}{c}{p}_1\\ p_2\end{array}\right].$$

(77)

Therefore, a necessary and sufficient condition for the solvability of Equation (74) is

$$\mathrm{rank}\left[\begin{array}{cc}L& p_2\end{array}\right]=\mathrm{rank}\left(L\right),$$

in which case, we can obtain x by solving $Lx=P_2$, and then compute y recursively. Furthermore, the upper bound on the degree of $\mathfrak{Y}$ in (75) are also given as follows:

Theorem 54 

([220]). For $\mathfrak{A}$ and $\mathfrak{B}$ given in (75), assume that

(1) 

$\mathfrak{A}$ and $\mathfrak{B}$ are relatively left prime;

(2) 

$\mathfrak{B}$ is nonsingular and satisfies that ${\mathfrak{B}}^{-1}$ is strictly proper;

(3) 

${\mathfrak{A}}_1\left(\lambda \right){\mathfrak{B}}_1^{-1}\left(\lambda \right)$ is the right coprime factorization of ${\mathfrak{B}}^{-1}\mathfrak{A}$, where ${\mathfrak{B}}_1\left(\lambda \right)$ is row reduced.

If Equation (75) is consistent, then it has a solution pair $\left(\mathfrak{X}\left(\lambda \right),\mathfrak{Y}\left(\lambda \right)\right)$ such that

$${deg}_{ri}\mathfrak{X}\left(\lambda \right)<{deg}_{ri}{\mathfrak{B}}_1\left(\lambda \right)\mathit{and}deg\mathfrak{Y}\left(\lambda \right)<deg{\mathfrak{A}}_1\left(\lambda \right),$$

where ${deg}_{ri}$ denotes the degree of the i-th row.

Remark 51. 

Żak [220] also noted that the number of equations in (77) can be further reduced by using additional information about A and B given in (74). For instance, if a row of B has a degree of zero, then the corresponding row of X is identically zero, so discard the relevant column and row of L given in (77).

6.6.5. By the Equivalence of Block Polynomial Matrices

Let $\mathbb{F}$ be a field. Building upon Theorem 1, Wimmer [222] investigated the constant solutions of the following polynomial matrix equation:

$$A\left(\lambda \right)X-YB\left(\lambda \right)=C\left(\lambda \right),$$

(78)

where $A\left(\lambda \right)\in {\mathbb{F}}^{m\times n}\left[\lambda \right]$, $B\left(\lambda \right)\in {\mathbb{F}}^{p\times k}\left[\lambda \right]$, and $C\left(\lambda \right)\in {\mathbb{F}}^{m\times k}\left[\lambda \right]$ are given.

Theorem 55 

(Lemma 2.1, [222]). Equation (78) has a constant solution pair

$$X\in {\mathbb{F}}^{n\times k}\mathit{and}Y\in {\mathbb{F}}^{m\times p}$$

if and only if there exist two nonsingular constant matrices

$$R\in {\mathbb{F}}^{(n+k)\times (n+k)}\mathit{and}S\in {\mathbb{F}}^{(m+p)\times (m+p)}$$

such that

$$\left[\begin{array}{cc}A\left(\lambda \right)& 0\\ 0& B\left(\lambda \right)\end{array}\right]R=S\left[\begin{array}{cc}A\left(\lambda \right)& C\left(\lambda \right)\\ 0& B\left(\lambda \right)\end{array}\right].$$

Remark 52. 

Theorem 55 also appeared as a lemma in the earlier article (Lemma 3, [223]), though no complete proof was provided.

Remark 53. 

Let $A_i\in {\mathbb{F}}^{m\times n}$, $B_i\in {\mathbb{F}}^{p\times k}$, and $C_i\in {\mathbb{F}}^{m\times k}$ for $i=1,2$. Using Theorem 55, Wimmer (Theorem 1.1, [222]) showed that the system of matrix equations

$$\left\{\begin{array}{c}{A}_{1}X-Y{B}_1=C_1,\hfill \\ A_{2}X-Y{B}_2=C_2,\hfill \end{array}\right.$$

has a solution pair $X\in {\mathbb{F}}^{n\times k}$ and $Y\in {\mathbb{F}}^{m\times p}$ if and only if there exist two nonsingular matrices $R\in {\mathbb{F}}^{(n+k)\times (n+k)}$ and $S\in {\mathbb{F}}^{(m+p)\times (m+p)}$ such that

$$S\left(\left[\begin{array}{cc}{A}_1& C_1\\ 0& B_1\end{array}\right]-\lambda \left[\begin{array}{cc}{A}_2& C_2\\ 0& B_2\end{array}\right]\right)=\left(\left[\begin{array}{cc}{A}_1& 0\\ 0& B_1\end{array}\right]-\lambda \left[\begin{array}{cc}{A}_2& 0\\ 0& B_2\end{array}\right]\right)R.$$

Obviously, this result is also a generalization of RET. A similar research idea is also introduced in Section 8.1.

6.6.6. By Jordan Systems of Polynomial Matrices

Let $A\in {\mathbb{C}}^{m\times m}\left[\lambda \right]$, $B\in {\mathbb{C}}^{n\times n}\left[\lambda \right]$, and $C\in {\mathbb{C}}^{m\times n}\left[\lambda \right]$ be such that $\det \left(A\right)\ne 0$ and $\det \left(B\right)\ne 0$. According to Jordan systems of polynomial matrices, Wimmer [224] discussed the solvability conditions for the polynomial matrix equations

$$AX-YB=C,$$

(79)

where $X,Y\in {\mathbb{C}}^{m\times n}\left[\lambda \right]$ are unknown. Let

$$\sigma \left(B\right)=\{\lambda \in \mathbb{C}\mid det(B\left(\lambda \right))=0\},$$

and let the elementary divisors corresponding to ${\lambda }_1\in \sigma \left(B\right)$ be

$${(\lambda -{\lambda }_1)}^{l_1},\dots ,{(\lambda -{\lambda }_1)}^{l_q},$$

where $l_1\ge \dots \ge l_q\ge 1$ and $l=l_1+\dots +l_q$ satisfying

$$det\left(B\right)={(\lambda -{\lambda }_1)}^{l}c\left(\lambda \right)\mathrm{and}c\left({\lambda }_1\right)\ne 0.$$

For $r\in {\mathbb{Z}}^{+}$, denote

$$N_r={\left[\begin{array}{cccc}0& 1& & \\ & \ddots & \ddots & \\ & & \ddots & 1\\ & & & 0\end{array}\right]}_{r\times r}.$$

Let the Jordan matrix of B associated to ${\lambda }_1$ be

$$J=\mathrm{diag}({\lambda }_{1}I-N_{l_1},\dots ,{\lambda }_{1}I-N_{l_q}).$$

(80)

Then, there exist $H\in {\mathbb{C}}^{n\times l}$ and $\widehat{H}\in {\mathbb{C}}^{n\times l}\left[\lambda \right]$ such that

$$BH=\widehat{H}(\lambda I-J),$$

where the columns of $\widehat{H}$ are $\mathbb{C}$-linearly independent (see [225]). Thus, H is called a right Jordan system of B corresponding to ${\lambda }_1$. Then, G is called a left Jordan system of A corresponding to ${\lambda }_1\in \sigma \left(A\right)$ if $G^T$ is a right Jordan system of $A^T$.

For ${\lambda }_1\in \sigma \left(A\right)\cap \sigma \left(B\right)$, let G and H be left and right Jordan systems of A and B corresponding to ${\lambda }_1$, respectively. Then, $(G,H)$ is called a pair of Jordan systems of $(A,B)$ corresponding to ${\lambda }_1$. According to (80), partition

$$H=\left[\begin{array}{cccc}{H}_1& H_2& \dots & H_q\end{array}\right]$$

with $H_j=[h_{j0},\dots ,h_{j,l_j-1}]$ for $j=1,\dots ,q$. Similarly, assume that the elementary divisors belonging to ${\lambda }_1\in \sigma \left(A\right)$ are

$${(\lambda -{\lambda }_1)}^{k_1},\dots ,{(\lambda -{\lambda }_1)}^{k_p},$$

where $k_1\ge \dots \ge k_p\ge 1$. Partition

$$G=\left[\begin{array}{c}{G}_1\\ ⋮\\ G_p\end{array}\right]\mathrm{with}{G}_i=\left[\begin{array}{c}{g}_{i0}\\ ⋮\\ g_{i,k_i-1}\end{array}\right]\mathrm{for}i=1,\dots ,p.$$

The k-th derivative of $C\left(\lambda \right)$ is denoted by $C^{\left(k\right)}\left(\lambda \right)$. We call that $(G,H)$ has the property $\left(\mathsf{\Sigma }\right)$ if

$$\sum _{\nu +\sigma +\tau =r_{ij}}{g}_{i\nu }{\displaystyle \frac{C^{\left(\sigma \right)}\left({\lambda }_1\right)}{\sigma !}}{h}_{j\tau }=0,$$

where $r_{ij}=0,1,\dots ,min(k_i,l_j)-1,i=1,\dots ,p$, and $j=1,\dots ,q$.

Theorem 56 

(Theorem 1.1, [224]). Let A, B, and C be given in (79). Then, the following are equivalent:

(1) 

Equation (79) is consistent;

(2) 

There exists a pair of Jordan systems $(G,H)$ of $(A,B)$ with property $\left(\mathsf{\Sigma }\right)$ for each $\lambda \in \sigma \left(A\right)\cap \sigma \left(B\right)$;

(3) 

All pairs of Jordan systems of $(A,B)$ have property $\left(\mathsf{\Sigma }\right)$ for each $\lambda \in \sigma \left(A\right)\cap \sigma \left(B\right)$.

Remark 54. 

Wimmer [224] pointed out that Theorem 56 can also be extended to matrices over the ring $\mathcal{O}\left(\mathbb{G}\right)$ of complex holomorphic functions in a domain $\mathbb{G}$.

6.6.7. By Linear Matrix Equations

Let $\mathbb{F}$ be a field. Assume that

$$A\left(\lambda \right)=\sum _{i=0}^m{A}_i{\lambda }^i,B\left(\lambda \right)=\sum _{i=0}^n{B}_i{\lambda }^i,\mathrm{and}C\left(\lambda \right)=\sum _{i=0}^k{C}_i{\lambda }^i,$$

(81)

where $A_i,B_i,C_i\in {\mathbb{F}}^{r\times r}$ satisfying $A_m\ne 0$, $B_n\ne 0$, and $C_k\ne 0$. Consider the polynomial matrix equation:

$$A\left(\lambda \right)X\left(\lambda \right)+Y\left(\lambda \right)B\left(\lambda \right)=C\left(\lambda \right),$$

(82)

where $X\left(\lambda \right),Y\left(\lambda \right)\in {\mathbb{F}}^{r\times r}\left[\lambda \right]$ are unknown.

Barnett [226] provided an equivalent condition for Equation (82) to have a unique solution pair $\left(X\right(\lambda ),Y(\lambda \left)\right)$ with

$$degX\left(\lambda \right)<n\mathrm{and}degY\left(\lambda \right)<m.$$

(83)

For $H\left(\lambda \right)={\sum }_{i=0}^m{H}_i{\lambda }^i$ with $H_m\ne 0$, we say that $H\left(\lambda \right)$ is regular if $det\left(H_m\right)\ne 0$.

Theorem 57 

([226]). Let $A\left(\lambda \right)$, $B\left(\lambda \right)$, and $C\left(\lambda \right)$ given in (81) be such that $A\left(\lambda \right)$ and $B\left(\lambda \right)$ are regular and that $degC\left(\lambda \right)\le n+m-1$. Then, Equation (82) has a unique solution pair $\left(X\right(\lambda ),Y(\lambda \left)\right)$ such that (83) if and only if $det\left(A\right(\lambda \left)\right)$ and $det\left(B\right(\lambda \left)\right)$ are relatively prime, (i.e., the greatest common divisor is a constant independent λ).

Remark 55. 

Note that the condition in Theorem 57, i.e., that $det\left(A\right(\lambda \left)\right)$ and $det\left(B\right(\lambda \left)\right)$ are relatively prime, implies that $A\left(\lambda \right)$ and $B\left(\lambda \right)$ are nonsingular (see [227]).

Subsequently, Feinstein and Bar-Ness [227] conducted further research on the solutions to Equation (82) with (83); such special solutions are also called the minimal solutions.

Theorem 58 

(Theorem II, [227]). Let $A\left(\lambda \right)$, $B\left(\lambda \right)$, and $C\left(\lambda \right)$ given in (81) be such that $A\left(\lambda \right)$ and $B\left(\lambda \right)$ are nonsingular and $degC\left(\lambda \right)\le n+m-1$. Assume that $A\left(\lambda \right)$ (or $B\left(\lambda \right)$) is regular. Then, Equation (82) has a unique minimal solution if and only if $det\left(A\right(\lambda \left)\right)$ and $det\left(B\right(\lambda \left)\right)$ are relatively prime.

Remark 56. 

In Theorem III of [227], Feinstein and Bar-Ness showed that, if Equation (82) has a minimal solution $\left(X\right(\lambda ),Y(\lambda \left)\right)$, then $\left(X\right(\lambda ),Y(\lambda \left)\right)$ is not unique if and only if both $A\left(\lambda \right)$ and $B\left(\lambda \right)$ are not regular.

Motivated by Theorem 57, Chen and Tian [228] proved that Equation (81) can be reduced to a linear matrix equation. For $A\left(\lambda \right)$ and $B\left(\lambda \right)$ given in (81), let

$$A_R=\left[\begin{array}{ccccc}0& 0& \dots & 0& -A_0\\ I& 0& \dots & 0& -A_1\\ 0& I& \dots & 0& -A_2\\ ⋮& ⋮& \ddots & ⋮& ⋮\\ 0& 0& \dots & I& -A_{m-1}\end{array}\right]\mathrm{and}{B}_L=\left[\begin{array}{ccccc}0& I& 0& \dots & 0\\ 0& 0& I& \dots & 0\\ ⋮& ⋮& ⋮& \ddots & ⋮\\ 0& 0& 0& \dots & I\\ -B_0& -B_1& -B_2& \dots & -B_{n-1}\end{array}\right].$$

Theorem 59 

(Lemma 3.1 and Theorems 1.1 and 1.3, [228]). Let $A\left(\lambda \right)$, $B\left(\lambda \right)$, and $C\left(\lambda \right)$ given in (81) be such that $A_m=B_n=I_r$, and let ${\mathbb{H}}_{C\left(\lambda \right)}$ be the set of all solutions $\left(X\right(\lambda ),Y(\lambda \left)\right)$ of Equation (82) with (83).

(1) 

Let $k<m+n$. If Equation (82) is solvable, then ${\mathbb{H}}_{C\left(\lambda \right)}\ne \varnothing$.

(2) 

Let $k<m$. There exists $X\left(\lambda \right)$ satisfying $(X\left(\lambda \right),Y\left(\lambda \right))\in {\mathbb{H}}_{C\left(\lambda \right)}$ if and only if

$$A_R^{n}Y+\sum _{i=0}^{n-1}{A}_R^{i}Y{B}_i=C,$$

(84)

where

$$Y\left(\lambda \right)=\sum _{i=0}^{m-1}{Y}_i{\lambda }^i,Y=\left[\begin{array}{c}{Y}_0\\ ⋮\\ Y_{m-1}\end{array}\right],C=\left[\begin{array}{c}{C}_0\\ ⋮\\ C_{m-1}\end{array}\right].$$

(3) 

Let $k<n$. There exists $Y\left(\lambda \right)$ satisfying $(X\left(\lambda \right),Y\left(\lambda \right))\in {\mathbb{H}}_{C\left(\lambda \right)}$ if and only if

$$X{B}_L^m+\sum _{i=0}^{n-1}{A}_{i}X{B}_L^i=\tilde{C},$$

(85)

where $X\left(\lambda \right)={\sum }_{i=0}^{n-1}{X}_i{\lambda }^i$, $X=\left[\begin{array}{ccc}{X}_0& \dots & X_{n-1}\end{array}\right]$, and $\tilde{C}=\left[\begin{array}{ccc}{C}_0& \dots & C_{n-1}\end{array}\right]$.

Remark 57. 

(1) 

For $A=\left[a_{ij}\right]\in {\mathbb{F}}^{p\times q}$, let

$$\mathrm{row}\left(A\right)=\left[\begin{array}{cccccccccc}{a}_{11}& \dots & a_{p1}& a_{12}& \dots & a_{p2}& \dots & a_{1q}& \dots & a_{pq}\end{array}\right].$$

and $\mathrm{vec}\left(A\right)=\mathrm{row}{\left(A\right)}^T$. Then,

$$\begin{array}{cc}\hfill \left(84\right)& \iff \left(I\otimes A_R^n+\sum _{i=0}^{n-1}{B}_i^T\otimes A_R^i\right)\mathrm{vec}\left(Y\right)=\mathrm{vec}\left(C\right),\hfill \\ \hfill \left(85\right)& \iff \mathrm{row}\left(X\right)\left(I\otimes B_L^m+\sum _{i=0}^{m-1}{A}_i\otimes B_L^i\right)=\mathrm{row}\left(\tilde{C}\right).\hfill \end{array}$$

(2) 

The explicit solutions to Equations (84) and (85) have been studied in [229,230], which also serve as a starting point of Section 6.7 in this paper.

(3) 

Moreover, Sheng and Tian [228] mentioned that Theorem 59 still holds when the field $\mathbb{F}$ is extended to a commutative ring with identity.

6.6.8. By Root Functions of Polynomial Matrices

Let $L_1^{n\times n}[a,b]$ denote the space of $n\times n$ matrix-valued functions that are Lebesgue integrable over the interval $[a,b]$. Let

$$\mathcal{B}\left(\lambda \right)=I_n+{\int }_{-\omega }^0{\mathrm{e}}^{\mathrm{i}\lambda t}b\left(t\right)\mathrm{d}t,\mathcal{D}\left(\lambda \right)=I_n+{\int }_0^{\omega }{\mathrm{e}}^{\mathrm{i}\lambda t}d\left(t\right)\mathrm{d}t,G\left(\lambda \right)={\int }_{-\omega }^{\omega }{\mathrm{e}}^{\mathrm{i}\lambda t}g\left(t\right)\mathrm{d}t,$$

where $b\in L_1^{n\times n}[-\omega ,0]$, $d\in L_1^{n\times n}[0,\omega ]$, and $g\in L_1^{n\times n}[-\omega ,\omega ]$. Consider the following linear entire matrix function equation

$$X\left(\lambda \right)\mathcal{B}\left(\lambda \right)+\mathcal{D}\left(\lambda \right)Y\left(\lambda \right)=G\left(\lambda \right),\lambda \in \mathbb{C},$$

(86)

where X and Y are unknown $n\times n$ matrix functions

$$X\left(\lambda \right)={\int }_0^{\omega }{\mathrm{e}}^{\mathrm{i}\lambda t}x\left(t\right)\mathrm{d}t\mathrm{and}Y\left(\lambda \right)={\int }_{-\omega }^0{\mathrm{e}}^{\mathrm{i}\lambda t}y\left(t\right)\mathrm{d}t$$

with $x\in L_1^{n\times n}[0,\omega ]$ and $y\in L_1^{n\times n}[-\omega ,0]$. Gohberg [231] proposed necessary and sufficient conditions for the solvability of Equation (86) using the root functions of the coefficients.

Definition 17 

([231,232]). Let an $n\times n$ matrix function $H\left(\lambda \right)$ be analytic at ${\lambda }_0\in \mathbb{C}$. A ${\mathbb{C}}^n$-valued function φ is called a root function of $H\left(\lambda \right)$ at ${\lambda }_0$ of order (at least) $k\in {\mathbb{Z}}^{+}$ if φ is analytic at ${\lambda }_0$, $\phi \left({\lambda }_0\right)\ne 0$, and $H\left(\lambda \right)\phi \left(\lambda \right)$ has a zero at ${\lambda }_0$ of order (at least) k.

Theorem 60 

(Theorem 1.1, [231]). Equation (86) is consistent if and only if for each common zero ${\lambda }_0$ of $det\left(\mathcal{B}\right(\lambda \left)\right)$ and $det\left(\mathcal{D}\right(\lambda \left)\right)$, if φ is a root function of $\mathcal{B}\left(\lambda \right)$ at ${\lambda }_0$ of order p and ψ is a root function of $\mathcal{D}{\left(\lambda \right)}^T$ at ${\lambda }_0$ of order q, then $\psi {\left(\lambda \right)}^{T}G\left(\lambda \right)\phi \left(\lambda \right)$ has a zero at ${\lambda }_0$ of order at least $min\{q,p\}$.

Let $H\left(\lambda \right)$ be an analytic $r\times r$ matrix function, and let ${\lambda }_0$ be a point in the domain of analyticity of $H\left(\lambda \right)$. If $det\left(H\left({\lambda }_0\right)\right)=0$, then ${\lambda }_0$ is called an eigenvalue of $H\left(\lambda \right)$. Let a ${\mathbb{C}}^r$-vector valued function $\varphi \left(\lambda \right)$ be analytic in a neighborhood of an eigenvalue ${\lambda }_0$ of $F\left(\lambda \right)$. If $\varphi \left({\lambda }_0\right)\ne 0$ and $F\left({\lambda }_0\right)\varphi \left({\lambda }_0\right)=0$, then $\varphi \left(\lambda \right)$ is called a right root function of $F\left(\lambda \right)$ at ${\lambda }_0$. The order (at least) k of the right root function $\varphi \left(\lambda \right)$ at ${\lambda }_0$ is the order (at least) k of ${\lambda }_0$ as a zero of the analytic function $F\left(\lambda \right)\varphi \left(\lambda \right)$. Similarly, left root functions can be defined (see [233]).

Utilizing the right and left root functions of polynomial matrices, Kaashoek and Lerer [233] then presented a discrete version of Theorem 60. Let

$$L\left(\lambda \right)=\sum _{j=0}^l{\lambda }^j{L}_j,M\left(\lambda \right)=\sum _{j=0}^m{\lambda }^j{M}_j,G\left(\lambda \right)=\sum _{j=0}^{\ell +m-1}{\lambda }^j{G}_j,$$

(87)

where $L_j,M_j,G_j\in {\mathbb{C}}^{r\times r}$ with $L_l\ne 0$ and $M_m\ne 0$. Consider the following polynomial matrix equation:

$$X\left(\lambda \right)L\left(\lambda \right)+M\left(\lambda \right)Y\left(\lambda \right)=G\left(\lambda \right),$$

(88)

where $X\left(\lambda \right)$ and $Y\left(\lambda \right)$ are unknown. Define

$$\begin{array}{cc}\hfill & \widehat{L}\left(\lambda \right)={\lambda }^{l}L\left({\lambda }^{-1}\right)=\sum _{j=0}^l{\lambda }^j{L}_{l-j},\widehat{M}\left(\lambda \right)={\lambda }^{m}M\left({\lambda }^{-1}\right)=\sum _{j=0}^m{\lambda }^j{M}_{m-j},\hfill \\ \hfill & \overline{G}\left(\lambda \right)={\lambda }^{l+m-1}G\left({\lambda }^{-1}\right)=\sum _{j=0}^{l+m-1}{\lambda }^j{G}_{l+m-1-j}.\hfill \end{array}$$

Theorem 61 

(Theorem 1.1, [233]). For $L\left(\lambda \right)$ and $M\left(\lambda \right)$ given in (87), assume that both $det\left(L\right(\lambda \left)\right)$ and $det\left(M\right(\lambda \left)\right)$ do not vanish identically. Then, Equation (88) has a solution pair $\left(X\right(\lambda ),Y(\lambda \left)\right)$ such that

$$degX\left(\lambda \right)\le m-1\mathit{and}degY\left(\lambda \right)\le l-1$$

if and only if both of the following two conditions hold:

(1) 

For each ${\lambda }_0\in \mathbb{C}$ satisfying $det\left(L\left({\lambda }_0\right)\right)=det\left(M\left({\lambda }_0\right)\right)=0$, if $f\left(\lambda \right)$ is a right root function of $L\left(\lambda \right)$ at ${\lambda }_0$ of order s and $h\left(\lambda \right)$ is a left root function of $M\left(\lambda \right)$ at ${\lambda }_0$ of order t, then $h\left(\lambda \right)G\left(\lambda \right)f\left(\lambda \right)$ has a zero at ${\lambda }_0$ of order at least $min\{s,t\}$;

(2) 

If $f_{\circ }\left(\lambda \right)$ is a right root function of $\widehat{L}\left(\lambda \right)$ at zero of order $s_{\circ }$ and $h_{\circ }\left(\lambda \right)$ is a left root function of $\widehat{M}\left(\lambda \right)$ at zero of order $t_{\circ }$, then $h_{\circ }\left(\lambda \right)\overline{G}\left(\lambda \right)f_{\circ }\left(\lambda \right)$ has a zero of order at least $min\{s_{\circ },t_{\circ }\}$.

Remark 58. 

Kaashoek and Lerer proved Theorem 61 by means of Theorem 1.2 of [233], which is a direct corollary of Theorems 3.2 and 4.1 of [234]. Although this proof strategy can be regarded as a discrete version of the proof of Theorem 60, in the discrete case, the common spectrum at infinity plays a crucial role, whereas in [231] there are no common root functions at infinity.

6.7. Sylvester-Polynomial-Conjugate Matrix Equations

For given matrices A, $B_i(i=1,\dots ,k)$, and C, the matrix equation

$$\sum _{i=0}^k{A}^{i}X{B}_i=C$$

has been thoroughly investigated for its important role in control theory (see [229,230,235,236,237]). Based on results in [230], Wu et al. [238] defined Sylvester sums and Kronecker maps, and used them to discuss the following matrix equation over $\mathbb{R}$:

$$\sum _{i=0}^{n_1}{A}_{i}X{F}^i=\sum _{i=0}^{n_2}{B}_{i}R{F}^i,$$

where X is unknown and other matrices are given over $\mathbb{R}$. Building on the work in [238], Wu et al. [239,240,241] established the framework of conjugate products and Sylvester-conjugate sums over $\mathbb{C}$. Specifically, for $A\in {\mathbb{C}}^{m\times n}$ and $k\in \mathbb{N}$, define

$$A^{*k}=\left\{\begin{array}{cc}A,\hfill & \mathrm{for}\mathrm{even}k,\hfill \\ \overline{A},\hfill & \mathrm{for}\mathrm{odd}k.\hfill \end{array}\right.$$

The conjugate product [239,241] of

$$A\left(\lambda \right)=\sum _{i=0}^m{A}_i{\lambda }^i\in {\mathbb{C}}^{p\times q}\left[\lambda \right]\mathrm{and}B\left(\lambda \right)=\sum _{j=0}^n{B}_j{\lambda }^j\in {\mathbb{C}}^{q\times r}\left[\lambda \right]$$

is defined as

$$A\left(\lambda \right)⊛B\left(\lambda \right)=\sum _{i=0}^m\sum _{j=0}^n{A}_i{B}_j^{*i}{\lambda }^{i+j}.$$

For $A\in {\mathbb{C}}^{n\times n}$ and $k\in \mathbb{N}$, define

$$A^{\overleftarrow{k}}=A^{k-2⌊{\displaystyle \frac{k}{2}}⌋}{\left(\overline{A}A\right)}^{⌊{\displaystyle \frac{k}{2}}⌋},$$

where $\lfloor ·\rfloor$ is the floor function (downward rounding). For $Z\in {\mathbb{C}}^{r\times p}$, $F\in {\mathbb{C}}^{p\times p}$, and

$$T\left(\lambda \right)=\sum _{i=0}^t{T}_i{\lambda }^i\in {\mathbb{C}}^{n\times r}\left[\lambda \right],$$

we define the Sylvester-conjugate sum [241] of $T\left(\lambda \right)$ and Z with respect to F by

$$T\left(\lambda \right)\stackrel{F}{⊞}Z=\sum _{i=0}^t{T}_i{Z}^{*i}{F}^{\overleftarrow{i}}.$$

(89)

Remark 59. 

Note that Lemma 7 in ref. [241] reveals an intriguing relationship between the conjugate product and the Sylvester-conjugate sum. Specifically, if $A\left(\lambda \right)\in {\mathbb{C}}^{l\times q}\left[\lambda \right]$, $B\left(\lambda \right)\in {\mathbb{C}}^{q\times r}\left[\lambda \right]$, $F\in {\mathbb{C}}^{p\times p}$, and $Z\in {\mathbb{C}}^{r\times p}$, then

$$\left(A\left(\lambda \right)⊛B\left(\lambda \right)\right)\stackrel{F}{⊞}Z=A\left(\lambda \right)\stackrel{F}{⊞}\left(B\left(\lambda \right)\stackrel{F}{⊞}Z\right).$$

On this basis, Wu et al. [241] investigated the following Sylvester-polynomial-conjugate matrix equation:

$$\sum _{i=0}^{{\varphi }_1}{A}_i{X}^{*i}{F}^{\overleftarrow{i}}+\sum _{j=0}^{{\varphi }_2}{B}_j{Y}^{*j}{F}^{\overleftarrow{j}}=\sum _{k=0}^{{\varphi }_3}{C}_k{R}^{*k}{F}^{\overleftarrow{k}},$$

(90)

where $A_i\in {\mathbb{C}}^{n\times n}(i=0,1,\dots ,{\varphi }_1)$, $B_j\in {\mathbb{C}}^{n\times r}(j=0,1,\dots ,{\varphi }_2)$, $C_k\in {\mathbb{C}}^{n\times m}(k=0,1,\dots ,{\varphi }_3)$, $R\in {\mathbb{C}}^{m\times p}$, and $F\in {\mathbb{C}}^{p\times p}$ are given, and $X\in {\mathbb{C}}^{n\times p}$ and $Y\in {\mathbb{C}}^{r\times p}$ are unknown. Denote

$$A\left(\lambda \right)=\sum _{i=0}^{{\varphi }_1}{A}_i{\lambda }^i,B\left(\lambda \right)=\sum _{i=0}^{{\varphi }_2}{B}_i{\lambda }^i,C\left(\lambda \right)=\sum _{i=0}^{{\varphi }_3}{C}_i{\lambda }^i,$$

(91)

where $A_i$, $B_i$, and $C_i$ are given in (90). Thus, by conjugate products and Sylvester-conjugate sums, Equation (90) can be directly rewritten as

$$A\left(\lambda \right)\stackrel{F}{⊞}X+B\left(\lambda \right)\stackrel{F}{⊞}Y=C\left(\lambda \right)\stackrel{F}{⊞}R.$$

(92)

Additionally, we say that $A\left(\lambda \right)\in {\mathbb{C}}^{n\times n}\left[\lambda \right]$ and $B\left(\lambda \right)\in {\mathbb{C}}^{n\times r}\left[\lambda \right]$ are left coprime [239] if all their greatest common left divisors are unimodular, which is also equivalent to the existence of $C\left(\lambda \right)\in {\mathbb{C}}^{n\times r}\left[\lambda \right]$ and $D\left(\lambda \right)\in {\mathbb{C}}^{r\times r}\left[\lambda \right]$ such that

$$A\left(\lambda \right)⊛C\left(\lambda \right)+B\left(\lambda \right)⊛D\left(\lambda \right)=I.$$

Theorem 62 

(Theorem 2, [241]). Assume that $A\left(\lambda \right)$ and $B\left(\lambda \right)$ given in (91) are left coprime. Suppose that the unimodular polynomial matrix $U\left(\lambda \right)\in {\mathbb{C}}^{(n+r)\times (n+r)}\left[\lambda \right]$ satisfies

$$\left[\begin{array}{cc}A\left(\lambda \right)& B\left(\lambda \right)\end{array}\right]⊛U\left(\lambda \right)=\left[\begin{array}{cc}I& 0\end{array}\right].$$

Then, all solutions of Equation (92) (or (90)) are

$$\left[\begin{array}{c}X\\ Y\end{array}\right]=\left(U\left(\lambda \right)⊛\left[\begin{array}{cc}C\left(\lambda \right)& 0\\ 0& I\end{array}\right]\right)\stackrel{F}{⊞}\left[\begin{array}{c}R\\ Z\end{array}\right],$$

(93)

where $Z\in {\mathbb{C}}^{r\times p}$ is arbitrary. Furthermore, if partition

$$U\left(\lambda \right)=\left[\begin{array}{cc}H\left(\lambda \right)& N\left(\lambda \right)\\ L\left(\lambda \right)& D\left(\lambda \right)\end{array}\right]$$

with $N\left(\lambda \right)\in {\mathbb{C}}^{n\times r}\left[\lambda \right]$ and $D\left(\lambda \right)\in {\mathbb{C}}^{r\times r}\left[\lambda \right]$, then (93) can be rewritten as

$$\left\{\begin{array}{c}X=\left(H\left(\lambda \right)⊛C\left(\lambda \right)\right)\stackrel{F}{⊞}R+N\left(\lambda \right)\stackrel{F}{⊞}Z,\hfill \\ Y=\left(L\left(\lambda \right)⊛C\left(\lambda \right)\right)\stackrel{F}{⊞}R+D\left(\lambda \right)\stackrel{F}{⊞}Z,\hfill \end{array}\right.$$

for arbitrary $Z\in {\mathbb{C}}^{r\times p}$.

Remark 60. 

(1) 

Theorem 9 in ref. [239] guarantees the existence of the polynomial matrix $U\left(\lambda \right)$ in Theorem 62.

(2) 

Taking

$${\varphi }_1=0,{\varphi }_2=1,B_0=0,B_1=I,\mathit{and}{\varphi }_3=0,$$

Equation (90) over $\mathbb{R}$ reduces to

$$AX+YB=C,$$

where $A=A_0$, $B=F$, and $C=C_{0}R$. Clearly, Theorem 62 is also a generalization of RET over $\mathbb{R}$.

(3) 

In Theorem 1 of [241], Wu et al. characterized the homogeneous case of Equation (90) more specifically via a pair of right coprime polynomial matrices. Moreover, in Remark 4 of [241], they utilized the same method to discuss a more general form of Equation (90), i.e.,

$$\sum _{k=1}^{\theta }\sum _{i=0}^{{\omega }_k}{A}_{ki}{X}_k{F}^{\overleftarrow{i}}=\sum _{j=0}^c{C}_{j}R{F}^{\overleftarrow{j}},$$

where $X_k(k=1,\dots ,\theta )$ are unknown and others are given.

(4) 

It can be observed that Lemmas 11 and 12 of [241] are crucial for proving Theorem 62 and Theorem 1 of [239]. Meanwhile, it should be noted that Lemmas 11 and 12 of [241] provide only necessary conditions for left and right coprimeness, respectively. Thus, we contend that exploring the converse problems of these two lemmas is interesting.

(5) 

Equation (90) generalizes a class of complex conjugate matrix equations (see [242,243,244,245,246,247,248]). For systematic research on complex conjugate matrix equations and their applications in discrete-time antilinear systems, refer to the monograph [249] by Wu and Zhang.

Inspired by [241], Mazurek [250] recently generalized Theorem 62 based on groupoids and vector spaces.

Theorem 63 

(Theorem 2, [250]). Let $M_{11},M_{12},M_{21},$ and $M_{22}$ be groupoids with binary operations commonly denoted by ⊕, and let $V_1$ and $V_2$ be finite-dimensional vector spaces over a field $\mathbb{F}$. Assume that for any $i,j,k\in \{1,2\}$, two operations

$$⊠:M_{ij}\times V_j\to V_i\mathit{and}\odot :M_{ij}\times M_{jk}\to M_{ik}$$

are given such that

(1) 

$(s\oplus t)⊠vs.=s⊠vs.+t⊠vs.$ for $i,j\in \{1,2\}$, $s,t\in M_{ij}$, and $vs.\in V_j$;

(2) 

$s⊠(ku+lv)=k(s⊠u)+l(s⊠v)$ for $i,j\in \{1,2\}$, $s\in M_{ij}$, $k,l\in \mathbb{F}$, and $u,vs.\in V_j$;

(3) 

$s⊠(t⊠v)=(s\odot t)⊠vs.$ for $i,j,k\in \{1,2\}$, $s\in M_{ij}$, $t\in M_{jk}$, and $vs.\in V_k$.

Suppose that for $a\in M_{11}$ and $b\in M_{12}$, there exist $p\in M_{11}$, $g\in M_{12}$, $d,q\in M_{21}$, and $h,w\in M_{22}$ such that

(i) 

$\left(\right(a\odot p)\oplus (b\odot q\left)\right)⊠vs.=vs.$ for any $vs.\in V_1$;

(ii) 

$\left(\right(d\odot g)\oplus (w\odot h\left)\right)⊠u=u$ for any $u\in V_2$,;

(iii) 

$\left(\right(a\odot g)\oplus (b\odot h\left)\right)⊠u=0$ for any $u\in V_2$.

Then, for $c\in V_1$, the all solutions of the equation

$$a⊠x+b⊠y=c$$

are

$$(x,y)=(p⊠c+g⊠z,q⊠c+h⊠z),$$

where $z\in V_2$ is arbitrary.

For a ring $\mathbb{F}$ with unity and a ring endomorphism $\sigma$ of $\mathbb{F}$, the skew polynomial ring $\mathbb{F}[\lambda ;\sigma ]$ is the set of polynomials over $\mathbb{F}$ in the indeterminate $\lambda$ with the usual addition and multiplication subject to $\lambda a=\sigma \left(a\right)\lambda$ for any $a\in \mathbb{F}$ (see [251]). Specifically, the multiplication in $\mathbb{F}[\lambda ;\sigma ]$ is given by

$$\left(\sum _{i=0}^n{a}_i{\lambda }^i\right)\left(\sum _{j=0}^m{b}_j{\lambda }^j\right)=\sum _{i=0}^n\sum _{j=0}^m{a}_i{\sigma }^i\left(b_j\right){\lambda }^{i+j}$$

(94)

for ${\sum }_{i=0}^n{a}_i{\lambda }^i,{\sum }_{j=0}^m{b}_j{\lambda }^j\in \mathbb{F}[\lambda ;\sigma ]$. The set of $m\times n$ matrices over the skew polynomial ring $\mathbb{F}[s;\sigma ]$ is denoted by ${\mathbb{F}}^{m\times n}[s;\sigma ]$. Assume that $\mathbb{F}$ is also a finite-dimensional vector space over a field $\mathbb{K}$, and put

$$\begin{array}{cc}\hfill M_{11}& ={\mathbb{F}}^{n\times n}[s;\sigma ],M_{12}={\mathbb{F}}^{n\times m}[s;\sigma ],M_{21}={\mathbb{F}}^{m\times n}[s;\sigma ],\hfill \\ \hfill M_{22}& ={\mathbb{F}}^{m\times m}[s;\sigma ],V_1={\mathbb{F}}^{n\times p},V_2={\mathbb{F}}^{m\times p},\hfill \end{array}$$

with the usual addition (denoted by ⊕) and the skew multiplication (94) (denoted by ⊙) of polynomial matrices. Then, applying Theorem 63, Mazurek obtained the relevant result (i.e., Theorem 3, [250]) for Equation

$$A\left(\lambda \right)⊠X+B\left(\lambda \right)⊠Y=C,$$

where $A\left(\lambda \right)\in {\mathbb{F}}^{n\times n}[\lambda ;\sigma ]$, $B\left(\lambda \right)\in {\mathbb{F}}^{n\times m}[\lambda ;\sigma ]$, and $C\in {\mathbb{F}}^{n\times p}$ are given, and $X\in {\mathbb{F}}^{n\times p}$ and $Y\in {\mathbb{F}}^{m\times p}$ are unknown.

Remark 61. 

In the proof of Theorem 1 in [250], Mazurek showed that Theorem 62 is an immediate corollary of Theorem 3 of [250].

Moreover, analogous to the conjugate product of complex matrices, Wu et al. [252] further defined the $\mathbf{j}$-conjugate product of quaternion matrices. The $\mathbf{j}$-conjugate of $A\in {\mathbb{H}}^{m\times n}$ is

$$\overleftrightarrow{A}=-\mathbf{j}A\mathbf{j}.$$

For $k\in {\mathbb{Z}}^{+}$, inductively define $A^{\circ k}=\overleftrightarrow{A^{\circ (k-1)}}\mathrm{with}{A}^{\circ 0}=A$. Then, the $\mathbf{j}$-conjugate product [252] of

$$A\left(\lambda \right)=\sum _{i=0}^m{A}_i{\lambda }^i\in {\mathbb{H}}^{p\times q}\left[\lambda \right]\mathrm{and}B\left(\lambda \right)=\sum _{j=0}^n{B}_j{\lambda }^j\in {\mathbb{H}}^{q\times r}\left[\lambda \right]$$

is defined as

$$A\left(\lambda \right)⊛B\left(\lambda \right)=\sum _{i=0}^m\sum _{j=0}^n{A}_i{B}_j^{\circ i}{\lambda }^{i+j}.$$

We say that $A\left(\lambda \right)\in {\mathbb{H}}^{n\times n}\left[\lambda \right]$ and $B\left(\lambda \right)\in {\mathbb{H}}^{n\times m}\left[\lambda \right]$ are ⊛-left coprime [250] if there exists a unimodular polynomial matrix $U\left(\lambda \right)\in {\mathbb{H}}^{(n+m)\times (n+m)}\left[\lambda \right]$ such that

$$\left[\begin{array}{cc}A\left(\lambda \right)& B\left(\lambda \right)\end{array}\right]⊛U\left(\lambda \right)=\left[\begin{array}{cc}I& 0\end{array}\right].$$

Let the map $\sigma$ be $\sigma :\mathbb{H}\to \mathbb{H}$ with $\sigma \left(h\right)=-\mathbf{j}h\mathbf{j}$ for $h\in \mathbb{H}$. Then, $\sigma$ is an automorphism on $\mathbb{H}$. So, the $\mathbf{j}$-conjugate product is the product of matrices over $\mathbb{H}[\lambda ;\sigma ]$. Similarly to (89), Mazurek [250] defined the Sylvester-$\mathbf{j}$-conjugate sum over $\mathbb{H}$,

$$T\left(\lambda \right)\stackrel{F}{⊞}V=T_{0}V+\sum _{m=1}^t{T}_m{\sigma }^m\left(V\right){\sigma }^{m-1}\left(F\right)\dots {\sigma }^1\left(F\right){\sigma }^0\left(F\right)$$

for $T\left(\lambda \right)={\sum }_{i=0}^t{T}_i{\lambda }^i\in {\mathbb{H}}^{n\times r}\left[\lambda \right]$, $V\in {\mathbb{H}}^{r\times p}$, and $F\in {\mathbb{H}}^{p\times p}$. Applying Theorem 3 of [250] to matrices over $\mathbb{H}[\lambda ;\sigma ]$ yields the following result immediately.

Theorem 64 

(Theorem 4, [250]). Let $A\left(\lambda \right)\in {\mathbb{H}}^{n\times n}\left[\lambda \right]$ and $B\left(\lambda \right)\in {\mathbb{H}}^{n\times m}\left[\lambda \right]$ be ⊛-left coprime. Then, there exist

$$P\left(\lambda \right)\in {\mathbb{H}}^{n\times n}\left[\lambda \right],G\left(\lambda \right)\in {\mathbb{H}}^{n\times m}\left[\lambda \right],D\left(\lambda \right),Q\left(\lambda \right)\in {\mathbb{H}}^{m\times n}\left[\lambda \right],H\left(\lambda \right),W\left(\lambda \right)\in {\mathbb{H}}^{m\times m}\left[\lambda \right]$$

such that

$$\begin{array}{cc}\hfill & A\left(\lambda \right)⊛P\left(\lambda \right)+B\left(\lambda \right)⊛Q\left(\lambda \right)=I_n,D\left(\lambda \right)⊛G\left(\lambda \right)+W\left(\lambda \right)⊛H\left(\lambda \right)=I_m,\hfill \\ \hfill & A\left(\lambda \right)⊛G\left(\lambda \right)+B\left(\lambda \right)⊛H\left(\lambda \right)=0.\hfill \end{array}$$

Moreover, given $F\in {\mathbb{H}}^{p\times p}$ and $C\in {\mathbb{H}}^{n\times p}$, the general solution of the matrix equation

$$A\left(\lambda \right)\stackrel{F}{⊞}X+B\left(\lambda \right)\stackrel{F}{⊞}Y=C$$

is

$$\left\{\begin{array}{c}X=P\left(\lambda \right)\stackrel{F}{⊞}C+G\left(\lambda \right)\stackrel{F}{⊞}Z,\hfill \\ Y=Q\left(\lambda \right)\stackrel{F}{⊞}C+H\left(\lambda \right)\stackrel{F}{⊞}Z,\hfill \end{array}\right.$$

where $Z\in {\mathbb{H}}^{m\times p}$ is arbitrary.

Remark 62. 

Theorem 64 not only generalizes Theorem 62 to $\mathbb{H}$, but also presents a more general result than the relevant results in [87,253,254,255,256,257,258,259,260,261].

6.8. Generalized Forms of GSE

In Section 4.3, we can see that the SVD plays an important role in solving Equation (1). Then, will the development of the SVD promote research on a more general form of Equation (1)? The work in [31,262,263] shows that it is affirmative.

De Moor and Zha [262] established GSVD of a finite number $k\in {\mathbb{Z}}^{+}$ of matrices over $\mathbb{C}$.

Theorem 65 

(GSVD for any k matrices (Theorem 1, [262])). Let

$$A_1\in {\mathbb{C}}^{n_0\times n_1},A_2\in {\mathbb{C}}^{n_1\times n_2},\dots ,A_{k-1}\in {\mathbb{C}}^{n_{k-2}\times n_{k-1}},\mathit{and}{A}_k\in {\mathbb{C}}^{n_{k-1}\times n_k}.$$

Then, there exist unitary matrices $U_1\in {\mathbb{C}}^{n_0\times n_0}$ and $V_k\in {\mathbb{C}}^{n_k\times n_k}$, and nonsingular matrices $X_j\in {\mathbb{C}}^{n_j\times n_j}(j=1,\dots ,k-1)$ such that

$$U_1^{*}{A}_1{X}_1={\mathsf{\Lambda }}_1,Z_1{A}_2{X}_2={\mathsf{\Lambda }}_2,\dots ,Z_{i-1}{A}_i{X}_i={\mathsf{\Lambda }}_i,\dots ,Z_{k-1}{A}_k{V}_k={\mathsf{\Lambda }}_k,$$

where $Z_j=X_j^{*}$ (or $=X_j^{-1}$) for $j=1,2,\dots ,k-1$ (i.e., both choices are always possible),

$$\begin{array}{cc}\hfill {\mathsf{\Lambda }}_j=& \left[\begin{array}{ccccccc}I& 0& 0& 0& \dots & 0& 0\\ 0& 0& 0& 0& \dots & 0& 0\\ 0& I& 0& 0& \dots & 0& 0\\ 0& 0& 0& 0& \dots & 0& 0\\ 0& 0& I& 0& \dots & 0& 0\\ 0& 0& 0& 0& \dots & 0& 0\\ \dots & \dots & \dots & \dots & \dots & \dots & \dots \\ 0& 0& 0& 0& \dots & I& 0\\ 0& 0& 0& 0& \dots & 0& 0\end{array}\right]\begin{array}{c}{r}_j^1\hfill \\ r_{j-1}^1-r_j^1\hfill \\ r_j^2\hfill \\ r_{j-1}^2-r_j^2\hfill \\ r_j^3\hfill \\ r_{j-1}^3-r_j^3\hfill \\ \dots \hfill \\ r_j^j\hfill \\ n_{j-1}-r_{j-1}-r_j^j\hfill \end{array}\hfill \\ \hfill & \begin{array}{ccccccc}{r}_j^1& r_j^2& r_j^3& r_j^4& \dots & r_j^j& n_j-r_j\end{array}\hfill \end{array}$$

($j=1,2,\dots ,k-1$) with $r_0=0$ and $r_j={\sum }_{i=1}^j{r}_j^i=\mathrm{rank}\left(A_j\right)$,

$$\begin{array}{cc}\hfill {\mathsf{\Lambda }}_k=& \left[\begin{array}{ccccccc}{\mathsf{\Lambda }}_k^1& 0& 0& 0& \dots & 0& 0\\ 0& 0& 0& 0& \dots & 0& 0\\ 0& {\mathsf{\Lambda }}_k^2& 0& 0& \dots & 0& 0\\ 0& 0& 0& 0& \dots & 0& 0\\ 0& 0& {\mathsf{\Lambda }}_k^3& 0& \dots & 0& 0\\ 0& 0& 0& 0& \dots & 0& 0\\ \dots & \dots & \dots & \dots & \dots & \dots & \dots \\ 0& 0& 0& 0& \dots & {\mathsf{\Lambda }}_k^k& 0\\ 0& 0& 0& 0& \dots & 0& 0\end{array}\right]\begin{array}{c}{r}_k^1\hfill \\ r_{k-1}^1-r_k^1\hfill \\ r_k^2\hfill \\ r_{k-1}^2-r_k^2\hfill \\ r_k^3\hfill \\ r_{k-1}^3-r_k^3\hfill \\ \dots \hfill \\ r_k^k\hfill \\ n_{k-1}-r_{k-1}-r_k^k\hfill \end{array}\hfill \\ \hfill & \begin{array}{ccccccc}{r}_k^1& r_k^2& r_k^3& r_k^4& \dots & r_k^k& n_k-r_k\end{array}\hfill \end{array}$$

(95)

with $r_k={\sum }_{i=1}^k{r}_k^i=\mathrm{rank}\left(A_k\right)$, and ${\mathsf{\Lambda }}_k^i\in {\mathbb{C}}^{r_k^i\times r_k^i}(i=1,2,\dots ,k)$ are diagonal with positive diagonal elements.

It is easy to see that there exists an elementary column transformation that turns ${\mathsf{\Lambda }}_k$ given in (95) into the matrix ${\mathsf{\Lambda }}_k^{\prime }$ consisting of only zero and identity matrix blocks. So, under the hypotheses of Theorem 65, there exist nonsingular matrices $U_1\in {\mathbb{C}}^{n_0\times n_0}$, $V_k^{\prime }\in {\mathbb{C}}^{n_k\times n_k}$, and $X_j\in {\mathbb{C}}^{n_j\times n_j}(j=1,\dots ,k-1)$ such that

$$U_1^{*}{A}_1{X}_1={\mathsf{\Lambda }}_1,Z_1{A}_2{X}_2={\mathsf{\Lambda }}_2,\dots ,Z_{i-1}{A}_i{X}_i={\mathsf{\Lambda }}_i,\dots ,Z_{k-1}{A}_k{V}_k^{\prime }={\mathsf{\Lambda }}_k^{\prime }.$$

(96)

In this case, ${\mathsf{\Lambda }}_1,\dots ,{\mathsf{\Lambda }}_{k-1}$, and ${\mathsf{\Lambda }}_k^{\prime }$ only have zero and identity blocks.

Following the idea analogous to that in (96), He (Lemma 2.1, [263]) considered the pure product singular value decomposition (PSVD) for four quaternion matrices, i.e.,

$$\left[\begin{array}{ccccc}{A}_1& & & & \\ I& A_2& & & \\ & I& A_3& & \\ & & I& A_4\end{array}\right],$$

where $A_1\in {\mathbb{H}}^{q_1\times q_2}$, $A_2\in {\mathbb{H}}^{q_2\times q_3}$, $A_3\in {\mathbb{H}}^{q_3\times q_4}$, and $A_4\in {\mathbb{H}}^{q_4\times q_5}$. Using the PSVD, He further investigated the system of generalized Sylvester matrix equations over $\mathbb{H}$:

$$\left\{\begin{array}{cc}\hfill X_1{A}_1-B_1{X}_2& =C_1,\hfill \\ \hfill X_2{A}_2-B_2{X}_3& =C_2,\hfill \\ \hfill X_3{A}_3-B_3{X}_4& =C_3,\hfill \\ \hfill X_4{A}_4-B_4{X}_5& =C_4,\hfill \end{array}\right.$$

where $X_1,X_2,\dots ,X_5$ are unknown (see Theorems 4.1 and 4.2, [263]).

Inspired by He’s aforementioned work, since the GSVD of an arbitrary number k of matrices has been established, can we then consider a system of k generalized Sylvester equations? That is to say, consider

$$\left\{\begin{array}{c}{X}_1{A}_1-B_1{X}_2=C_1,\hfill \\ X_2{A}_2-B_2{X}_3=C_2,\hfill \\ ⋮\hfill \\ X_k{A}_k-B_k{X}_{k+1}=C_k,\hfill \end{array}\right.$$

(97)

where $k\in {\mathbb{Z}}^{+}$, and $X_1,X_2,\dots ,X_{k+1}$ are unknown.

To answer this question, let us first take a look at the work of He et al. in [31]. In terms of Theorem 65, He et al. in Theorem 2.2 of [31] gave the simultaneous decomposition of fifteen matrices over $\mathbb{C}$, i.e.,

$$\left[\begin{array}{ccccccc}& B_1& & & & & \\ A_1& E_1& C_1& & & & \\ & D_1& & B_2& & & \\ & & A_2& E_2& C_2& \\ & & & D_2& & B_3& \\ & & & & A_3& E_3& C_3\\ & & & & & D_3\end{array}\right],$$

where $A_i,B_i,C_i,D_i,E_i(i=1,2,3)$ are given matrices over $\mathbb{C}$ with appropriate orders. By this simultaneous decomposition of fifteen matrices, they further studied the following system of complex matrix equations:

$$\left\{\begin{array}{c}{A}_1{X}_1{B}_1+C_1{X}_2{D}_1=E_1,\hfill \\ A_2{X}_2{B}_2+C_2{X}_3{D}_2=E_2,\hfill \\ A_3{X}_3{B}_3+C_3{X}_4{D}_3=E_3,\hfill \end{array}\right.$$

where $X_1,\dots ,X_4$ are unknown (see Theorems 3.1 and 3.6, [31]). Interestingly, they also demonstrated the simultaneous decomposition of $5k$ matrices by the same means, i.e.,

$$\left[\begin{array}{ccccccccc}& B_1& & & & & & & \\ A_1& E_1& C_1& & & & & & \\ & D_1& & B_2& & & & & \\ & & A_2& E_2& C_2& & & \\ & & & D_2& & \ddots & & \\ & & & & \ddots & \ddots & & & \\ & & & & & & & B_k& \\ & & & & & & A_k& E_k& C_k\\ & & & & & & & D_k\end{array}\right],$$

where $k\in {\mathbb{Z}}^{+}$, and $A_i,B_i,C_i,D_i,E_i(i=1,2,\dots ,k)$ are given matrices over $\mathbb{C}$ with appropriate orders (see Theorem 4.1, [31]). Next, it is natural to consider the system

$$\left\{\begin{array}{c}{A}_1{X}_1{B}_1+C_1{X}_2{D}_1=E_1,\hfill \\ A_2{X}_2{B}_2+C_2{X}_3{D}_2=E_2,\hfill \\ ⋮\hfill \\ A_k{X}_k{B}_k+C_k{X}_{k+1}{D}_k=E_k,\hfill \end{array}\right.$$

(98)

where $X_1,X_2,\dots ,X_{k+1}$ are unknown. Obviously, Problem (97) is a special case of the system (98). However, only one conjecture on the solvability conditions of the system (98) was presented, i.e., (Conjecture 4.2, [31]). The conjecture was not officially solved until 2025 (see Corollary 4.4, [34]). Moreover, in Theorem 2.1 of [34], He et al. further investigated

$$\left(98\right)\mathrm{subject}\mathrm{to}\left\{\begin{array}{c}{G}_1{X}_1=P_1,G_2{X}_2=P_2,\dots ,G_{k+1}{X}_{k+1}=P_{k+1},\hfill \\ X_1{H}_1=Q_1,X_2{H}_2=Q_2,\dots ,X_{k+1}{H}_{k+1}=Q_{k+1},\hfill \end{array}\right.$$

(99)

where $X_1,X_2,\dots ,X_{k+1}$ are unknown. Furthermore, Xie et al. [264] studied the following system over $\mathbb{H}$:

$$\left\{\begin{array}{c}{A}_1{X}_1+Y_1{B}_1+C_1{Z}_1{D}_1+F_1{Z}_2{G}_1=E_1,\hfill \\ A_2{X}_2+Y_2{B}_2+C_2{Z}_2{D}_2+F_2{Z}_3{G}_2=E_2,\hfill \\ A_3{X}_3+Y_3{B}_3+C_3{Z}_3{D}_3+F_3{Z}_4{G}_3=E_3,\hfill \end{array}\right.$$

(100)

where $X_i,Y_i,Z_i(i=1,2,3)$, and $Z_4$ are unknown.

Up to this point, we can observe that the systems (99) and (100) encompass most of the current formal generalizations of Equation (1) without considering differences in number sets, such as [121,265,266,267,268,269,270,271].

Remark 63. 

However, there are currently no published studies on the more generalized system as follows:

$$\begin{array}{cc}\hfill & \left\{\begin{array}{c}{\displaystyle \sum _{j=1}^l}{A}_{1,j}{X}_{1j}{B}_{1,j}+A_{1,l+1}{X}_{1,l+1}{B}_{1,l+1}+A_{1,l+2}{X}_{2,l+1}{B}_{1,l+2}=C_1,\hfill \\ {\displaystyle \sum _{j=1}^l}{A}_{2,j}{X}_{2,j}{B}_{2,j}+A_{2,l+1}{X}_{2,l+1}{B}_{2,l+1}+A_{2,l+2}{X}_{3,l+1}{B}_{2,l+2}=C_2,\hfill \\ ⋮\hfill \\ {\displaystyle \sum _{j=1}^l}{A}_{i,j}{X}_{i,j}{B}_{i,j}+A_{i,l+1}{X}_{i,l+1}{B}_{i,l+1}+A_{i,l+2}{X}_{i+1,l+1}{B}_{i,l+2}=C_i,\hfill \\ ⋮\hfill \\ {\displaystyle \sum _{j=1}^l}{A}_{k,j}{X}_{k,j}{B}_{k,j}+A_{k,l+1}{X}_{k,l+1}{B}_{k,l+1}+A_{k,l+2}{X}_{k+1,l+1}{B}_{k,l+2}=C_k,\hfill \end{array}\right.\hfill \\ \hfill \mathit{subject}\mathit{to}& \left\{\begin{array}{c}{G}_{i,j}{X}_{i,j}=P_{i,j},X_{i,j}{H}_{i,j}=Q_{i,j}\mathit{for}1\le i\le k,1\le j\le l+1,\hfill \\ G_{k+1,l+1}{X}_{k+1,l+1}=P_{k+1,l+1},X_{k+1,l+1}{H}_{k+1,l+1}=Q_{k+1,l+1},\hfill \end{array}\right.\hfill \end{array}$$

(101)

where $l,k\in {\mathbb{Z}}^{+},$$A_{i,j},B_{i,j},C_i$$(1\le i\le k,1\le j\le l+2)$, $G_{i,j},P_{i,j},H_{i,j},Q_{i,j}$$(1\le i\le k,1\le j\le l+1)$, $G_{k+1,l+1},P_{k+1,l+1},H_{k+1,l+1}$, and $Q_{k+1,l+1}$ are given, and $X_{i,j}$$(1\le i\le k,1\le j\le l+1)$ and $X_{k+1,l+1}$ are unknown. So, this is also an interesting problem.

Remark 64. 

Through the continuous research on the generalizations of Equation (1), it can be found that the GSVD gradually becomes ineffective, while the tool of the generalized inverse has always been an effective method. However, we also find that the generalized inverse theory has little success in studying the systems of equations that simultaneously couple multiple (≥2) unknown matrices.

For instance, for given matrices A, B, and C, the system

$$\left\{\begin{array}{c}AX-YB=C,\hfill \\ X-Y=0,\hfill \end{array}\right.$$

(102)

is consistent if and only if the Sylvester equation

$$AX-XB=C$$

(103)

is consistent. At present, there are almost no articles that directly represent the general solution of the Sylvester equation using only the generalized inverses of the coefficient matrices. However, it can be intuitively observed that, using the Kronecker product, SVD, or STP to discuss the Sylvester equation is a feasible option.

It is noted that Liu put forward an open problem in [40], that is, to study the equivalent conditions for the solvability of the system of matrix equations over $\mathbb{C}$:

$$\left\{\begin{array}{c}{A}_{1}X+Y{B}_1=C_1,\hfill \\ A_{2}X+Y{B}_2=C_2,\hfill \end{array}\right.$$

(104)

where X and Y are unknown. Clearly, the system (104) is a generalization of both Equation (5) and the system (102). Later, using the rank equalities, Wang et al. [114] solved this problem over $\mathbb{H}$ under the certain condition, i.e.,

$$\mathrm{rank}\left[\begin{array}{cc}{A}_1& A_2\end{array}\right]=\mathrm{rank}\left(A_1\right)+\mathrm{rank}\left(A_2\right)\mathit{and}\mathrm{rank}\left[\begin{array}{c}{B}_1\\ B_2\end{array}\right]=\mathrm{rank}\left(B_1\right)+\mathrm{rank}\left(B_2\right).$$

(105)

This wonderful work is illuminating for completely solving Equation (103) and the system (104) by using rank equalities or the generalized inverses.

Theorem 66 

(Theorem 2.8, [114]). Let $A_1,A_2\in {\mathbb{H}}^{m\times p},B_1,B_2\in {\mathbb{H}}^{q\times n},$ and $C_1,C_2\in {\mathbb{H}}^{m\times n}$ be such that every matrix equation in system (104) is consistent. If (105) holds, then system (104) is consistent if and only if

$$\begin{array}{cc}\hfill \mathrm{rank}\left[\begin{array}{cc}{B}_1& 0\\ B_2& 0\\ -C_1& A_1\\ C_2& A_2\end{array}\right]& =\mathrm{rank}\left[\begin{array}{c}{A}_1\\ A_2\end{array}\right]+\mathrm{rank}\left[\begin{array}{c}{B}_1\\ B_2\end{array}\right],\hfill \\ \hfill \mathrm{rank}\left[\begin{array}{cccc}{A}_1& A_2& -C_1& C_2\\ 0& 0& B_1& B_2\end{array}\right]& =\mathrm{rank}\left[\begin{array}{cc}{A}_1& A_2\end{array}\right]+\mathrm{rank}\left[\begin{array}{cc}{B}_1& B_2\end{array}\right],\hfill \end{array}$$

$$\begin{array}{cc}\hfill \mathrm{rank}\left[\begin{array}{ccc}0& B_1& B_2\\ A_1& 0& 0\\ A_2& 0& F\end{array}\right]& =\mathrm{rank}\left[\begin{array}{c}{A}_1\\ A_2\end{array}\right]+\mathrm{rank}\left[\begin{array}{cc}{B}_1& B_2\end{array}\right],\hfill \\ \hfill \mathrm{rank}\left[\begin{array}{ccc}0& B_1& B_2\\ A_1& 0& 0\\ A_2& 0& \widehat{F}\end{array}\right]& =\mathrm{rank}\left[\begin{array}{c}{A}_1\\ A_2\end{array}\right]+\mathrm{rank}\left[\begin{array}{cc}{B}_1& B_2\end{array}\right],\hfill \end{array}$$

where

$$\begin{array}{cc}\hfill F& =A_1(A_2^{(1,2)}{C}_2-A_1^{(1,2)}{C}_1){\left[\begin{array}{c}{B}_1\\ -B_2\end{array}\right]}^{(1,2)}\left[\begin{array}{c}{B}_1\\ -B_2\end{array}\right]+\Omega B_1,\hfill \\ \hfill \widehat{F}& =A_2(A_2^{(1,2)}{C}_2-A_1^{(1,2)}{C}_1){\left[\begin{array}{c}{B}_1\\ -B_2\end{array}\right]}^{(1,2)}\left[\begin{array}{c}{B}_1\\ -B_2\end{array}\right]+\Omega B_2,\hfill \\ \hfill \Omega & =\left[\begin{array}{cc}-A_1& A_2\end{array}\right]{\left[\begin{array}{cc}-A_1& A_2\end{array}\right]}^{(1,2)}\left(R_{A_2}{C}_2{B}_2^{(1,2)}-R_{A_1}{C}_1{B}_1^{(1,2)}\right)\hfill \end{array}$$

with $R_{A_1}=I-A_1{A}_1^{(1,2)}$.

Remark 65. 

For more research on Equation (104), please refer to [222,272].

7. Iterative Algorithms

Analytical solutions of GSE over various algebras have been introduced in Section 5 and Section 6. However, in practical applications, challenges such as high computational complexity, stability, and robustness issues often arise. Therefore, investigating numerical solutions to GSE is imperative. In this section, we primarily present the relevant results on the numerical solutions of GSE. First, several iterative algorithms that directly address the GSE are proposed.

In 1984, Ziętak [96] proposed an algorithm (Algorithm 1) for computing the $l_p$-solution of Equation (5) over $\mathbb{R}$, using the equivalence between items (1) and (3) of Theorem 17.

Theorem 67. 

Let $\left\{X_t\right\}$ and $\left\{Y_t\right\}$ be generalized by Algorithm 1, and let $1<p<\infty$. Then, the sequence $\{R_t=A{X}_t+Y_{t}B\}$ is convergent. Moreover, if $\widehat{X}\in {\mathbb{R}}^{r\times n}$ and $\widehat{Y}\in {\mathbb{R}}^{m\times s}$ satisfy $\widehat{R}=A\widehat{X}+\widehat{Y}B$, where

$$\widehat{R}=\underset{t\to \infty }{lim}{R}_t,$$

then $\widehat{X}$ and $\widehat{Y}$ are the $l_p$-solution of Equation (5).

Remark 66. 

When $p=2$, Ziętak (Theorem 3.3, [96]) showed that the iterative process in Algorithm 1 ends after two iterations for any arbitrary initial matrix $Y_0$, i.e., matrices $X_1$ and $Y_1$ are the $l_2$-solution of Equation (5).

Algorithm 1 Algorithm [96] for the $l_p$-solution of Equation (5) over $\mathbb{R}$

Require:  Given matrices $A\in {\mathbb{R}}^{m\times r}$, $B\in {\mathbb{R}}^{s\times n}$, and $C\in {\mathbb{R}}^{m\times n}$; Initial matrices: $Y_0\in {\mathbb{R}}^{m\times s}$; Ensure:  Sequences: $\{X_t\in {\mathbb{R}}^{r\times n}\}$ and $\{Y_t\in {\mathbb{R}}^{m\times s}\}$; 1: for $t=1,2,\dots$  do 2:     for $j=1,2,\dots ,n$ do 3:           Let $c_j$ and $b_j$ be j-th columns of C and B, respectively; 4:           Solve the $l_p$-solution $x_j^{\left(t\right)}$ of the linear system $Ax=c_j-Y_{t-1}{b}_j$; 5:           Set the j-th column of $X_t$ as $x_j^{\left(t\right)}$; 6:     end for 7:     for $i=1,2,\dots ,m$ do 8:           Let $d_i$ and $a_i$ be the i-th columns of $C^T$ and $A^T$, respectively; 9:           Solve the $l_p$-solution $y_i^{\left(t\right)}$ of the linear system $B^{T}y=d_i-X_t^T{a}_i$; 10:         Set the i-th column of $Y_t^T$ as $y_i^{\left(t\right)}$; 11:     end for 12:     Compute ${\gamma }_{2t-1}={\parallel A{X}_t+Y_{t-1}B-C\parallel }_p$; 13:     Compute ${\gamma }_{2t}={\parallel A{X}_t+Y_{t}B-C\parallel }_p$; 14:     if the sequence $\left\{{\gamma }_t\right\}$ is not decreasing then 15:           Break 16:     end if 17: end for

In 1985, analogous to Algorithm R [273] for a nonlinear matrix equation, Ziętak [95] devised Algorithm T (Algorithm 2). Based on this algorithm, Ziętak [95] further discussed the Chebyshev solution of Equation (5) under the conditions (29) and (28).

Algorithm 2 Algorithm T [95] for the Chebyshev Solution of Equation (5) over $\mathbb{R}$

Require:  Matrices $A\in {\mathbb{R}}^{m\times r}$, $B\in {\mathbb{R}}^{s\times n}$, and $C\in {\mathbb{R}}^{m\times n}$; Initial matrices $X_0\in {\mathbb{R}}^{r\times n}$ and $Y_0\in {\mathbb{R}}^{m\times s}$; Tolerance $ϵ>0$ (for termination criterion); Ensure:  Convergent sequences: $\{X_k\in {\mathbb{R}}^{r\times n}\}$ and $\{Y_k\in {\mathbb{R}}^{m\times s}\}$; 1: Initial residual matrix: $R_0=A{X}_0+Y_{0}B-C$; 2: Initial residual norm: ${\gamma }_0={\parallel R_0\parallel }_{\infty }$; 3: Set iteration index: $k\leftarrow 0$; 4: Termination criterion: $|{\gamma }_{2k+1}-{\gamma }_{2k}|<ϵ$; 5: while Termination criterion not satisfied do 6:       $R_{2k}=A{X}_k+Y_{k}B-C$; 7:       ${\gamma }_{2k}={\parallel R_{2k}\parallel }_{\infty }$; 8:       for $j=1$ to n do 9:           Extract the j-th column of $R_{2k}$: $r_j^{\left(2k\right)}$; 10:         Solve the Chebyshev solution $h_j^{\left(k\right)}$ of $Ah=-r_j^{\left(2k\right)}$; 11:         Set the j-th column of $X_{k+1}$ as $x_j^{(k+1)}=x_j^{\left(k\right)}+h_j^{\left(k\right)}$; 12:       end for 13:       $R_{2k+1}=A{X}_{k+1}+Y_{k}B-C$; 14:       ${\gamma }_{2k+1}={\parallel R_{2k+1}\parallel }_{\infty }$; 15:       for $i=1$ to m do 16:           Extract the i-th column of $R_{2k+1}^T$: $s_i^{(2k+1)}$; 17:           Solve the Chebyshev solution $g_i^{\left(k\right)}$ of $B^{T}g=-s_i^{(2k+1)}$; 18:           Set the i-th column of $Y_{k+1}^T$ as $y_i^{(k+1)}=y_i^{\left(k\right)}+g_i^{\left(k\right)}$; 19:       end for 20:       $k\leftarrow k+1$ 21: end while

Remark 67. 

From Algorithm 2, it follows that $x_j^{(k+1)}$ are the Chebyshev solutions of

$$Ax=c_j-Y_k{b}_j(j=1,\dots ,n)$$

and $y_i^{(k+1)}$ are the Chebyshev solutions of

$$B^{T}y=d_i-X_{k+1}^T{a}_i(i=1,\dots ,m),$$

where $c_j,b_j,d_i$, and $a_i$ are the columns of $C,B,C^T$ and $A^T$, respectively. Moreover, one can derive that the sequence $\left\{{\gamma }_l\right\}$ given by Algorithm 2 is convergent.

Theorem 68 

(Theorems 5.2 and 5.3, [95]). Let A, B, and C be given in Equation (5).

(1) 

Suppose that (29) is satisfied. If

$$Z_{*}=A{X}_{*}+Y_{*}B$$

is a cluster point of the sequence $\left\{Z_{2k}\right\}$ generated by Algorithm 2, then $X_{*}$ and $Y_{*}$ are a Chebyshev solution of Equation (5)

(2) 

Suppose that (28) is satisfied and matrix A satisfies Haar’s condition, i.e., $det\left(A_i\right)\ne 0$ for $i=1,\dots ,m$, where $A_i$ is the matrix obtained from A by deletion of the i-th row. Then, the matrices $X_1$ and $Y_1$ generated by Algorithm 2 are the Chebyshev solution of Equation (5) for arbitrary initial matrices $X_0$ and $Y_0$.

In 2008, using Kronecker products, Yang and Huang [274] derived the normwise backward errors of the approximate solutions of Equation (5) over $\mathbb{R}$, as well as their upper and lower bounds.

Definition 18 

([275]). Let $A\in {\mathbb{R}}^{m\times m}$, $B\in {\mathbb{R}}^{n\times n}$, and $C\in {\mathbb{R}}^{m\times n}$, and let $\tilde{X},\tilde{Y}\in {\mathbb{R}}^{m\times n}$ be a numerical solution of Equation (5). Put

$$\begin{array}{cc}\hfill & \eta =\left\{(\Delta A,\Delta B,\Delta C)\mid (A+\Delta A)\tilde{X}+\tilde{Y}(B+\Delta B)=C+\Delta C\right\}\hfill \\ \hfill \mathit{and}& \eta (\tilde{X},\tilde{Y})=\underset{(\Delta A,\Delta B,\Delta C)\in \eta }{min}{∥\left[{\displaystyle \frac{1}{{\theta }_1}}\Delta A,{\displaystyle \frac{1}{{\theta }_2}}\Delta B,{\displaystyle \frac{1}{{\theta }_3}}\Delta C\right]∥}_F,\hfill \end{array}$$

where ${\theta }_1,{\theta }_2,{\theta }_3>0$ are parameters. Then, $\eta (\tilde{X},\tilde{Y})$ is referred to as the relative backward error if ${\theta }_1={\parallel A\parallel }_F$, ${\theta }_2={\parallel B\parallel }_F$, ${\theta }_3={\parallel C\parallel }_F$; it is termed the absolute backward error if ${\theta }_1={\theta }_2={\theta }_3=1$.

Theorem 69 

(Theorem 1, [274]). Under the hypotheses Definition 18, let

$$R=C-A\tilde{X}-\tilde{Y}B\mathit{and}T=\left[{\theta }_1({\tilde{X}}^T\otimes I_m),{\theta }_2(I_n\otimes \tilde{Y}),-{\theta }_3(I_n\otimes I_m)\right].$$

Then

$${\displaystyle \frac{{\parallel R\parallel }_F}{\sqrt{n{\theta }_3^2+{\theta }_1^2\parallel \tilde{X}{\parallel }_F^2+{\theta }_2^2{\parallel \tilde{Y}\parallel }_F^2}}}\le \eta (\tilde{X},\tilde{Y})={\parallel T^{†}\mathrm{Vec}\left(R\right)\parallel }_2\le {\displaystyle \frac{{\parallel R\parallel }_F}{\sqrt{{\theta }_3^2+{\theta }_1^2{\lambda }_n\left({\tilde{X}}^T\tilde{X}\right)+{\theta }_1^2{\lambda }_m\left(\tilde{Y}{\tilde{Y}}^T\right)}}},$$

where $\mathrm{Vec}(·)$ denotes the column-wise vectorization of a matrix, and ${\lambda }_n(·)$ denotes the largest eigenvalue of a $n\times n$ positive semidefinite matrix.

In 2011, Li et al. [276] extended the classical conjugate gradient least-squares algorithm (abbreviated as CGLSA) to compute the optimal solution of Equation (5) over $\mathbb{R}$ with symmetric pattern constraints, i.e., consider

$$\underset{X\in {\mathbb{SR}}^{m\times m},Y\in {\mathbb{SR}}^{n\times n}}{min}{\parallel AX+YB-C\parallel }_F,$$

(106)

where $A,B,C\in {\mathbb{R}}^{n\times m}$, and ${\mathbb{SR}}^{m\times m}$ denotes the set of all $m\times m$ real symmetric matrices.

Theorem 70 

(Theorems 1 and 2, [276]). Let $A,B,C\in {\mathbb{R}}^{n\times m}$.

(1) 

For any arbitrary initial matrices $X^{\left(0\right)}\in {\mathbb{SR}}^{m\times m}$ and $Y^{\left(0\right)}\in {\mathbb{SR}}^{n\times n}$, the matrix sequence $\left\{X^{\left(k\right)}\right\}$ and $\left\{Y^{\left(k\right)}\right\}$ generated by Algorithm 3 will converge to a solution of Problem (106) with finitely many steps in exact arithmetic.

(2) 

Let

$$\mathbb{S}=\left\{[X,Y]|X={\displaystyle \frac{1}{2}}(A^{T}H+H^{T}A)\mathit{and}Y={\displaystyle \frac{1}{2}}(H{B}^T+B{H}^T)\right\}$$

for arbitrary $H\in {\mathbb{R}}^{n\times m}$. If $[X^{\left(0\right)},Y^{\left(0\right)}]\in \mathbb{S}$, then by Algorithm 3 we can obtain the unique least norm solution of Problem (106).

Remark 68. 

Under the conditions of Theorem 70, if matrix pair $[X,Y]\in \mathbb{S}$ is a solution of Problem (106), then it is the unique least norm solution. Additionally, Li et al. (Theorem 3, [276]) characterized the minimization property of Algorithm 3, which ensures that the algorithm converges smoothly.

The classic alternating direction method (abbreviated as ADM) is an extension of the augmented Lagrangian multiplier method [277]. In 2017, inspired by [278], Ke and Ma [279] applied ADM to solve the nonnegative solutions of Equation (5) over $\mathbb{R}$, i.e.,

$$\mathrm{find}X(\ge 0)\in {\mathbb{R}}^{s\times n}\mathrm{and}Y(\ge 0)\in {\mathbb{R}}^{m\times l}\mathrm{s}.\mathrm{t}.AX+YB=C,$$

where $A\in {\mathbb{R}}^{m\times s}$, $B\in {\mathbb{R}}^{l\times n}$, and $C\in {\mathbb{R}}^{m\times n}$. Note that Problem (107) is equivalent to the following quadratic programming problem:

$$\underset{\begin{array}{c}X\in {\mathbb{R}}^{s\times n},Y\in {\mathbb{R}}^{m\times l}\end{array}}{min}{\displaystyle \frac{1}{2}}{\parallel AX+YB-C\parallel }_F^2\mathrm{s}.\mathrm{t}.X\ge 0\mathrm{and}Y\ge 0,$$

which is further equivalent to

$$\underset{X,U\in {\mathbb{R}}^{s\times n},Y,V\in {\mathbb{R}}^{m\times l}}{min}{\displaystyle \frac{1}{2}}{\parallel AX+YB-C\parallel }_F^2\mathrm{s}.\mathrm{t}.X=U,Y=V,U\ge 0,\mathrm{and}V\ge 0.$$

(107)

For $A=\left(a_{ij}\right)\in {\mathbb{R}}^{m\times n}$, let ${\mathcal{P}}_{+}\left(A\right)=\left[b_{ij}\right]\in {\mathbb{R}}^{m\times n}$, where $b_{ij}=max\{0,a_{ij}\}$.

Definition 19 

([279]). A point $(X,Y,U,V)$ satisfies the KKT conditions for Problem (107), i.e., there exist the matrices Λ and Π such that

$$\begin{array}{cc}\hfill & A^T(AX+YB-C)+\mathsf{\Lambda }=0,(AX+YB-C)B^T+\mathsf{\Pi }=0,X-U=0,\hfill \\ \hfill & Y-V=0,\mathsf{\Lambda }\le 0\le U,\mathsf{\Lambda }\odot U=0,\mathsf{\Pi }\le 0\le V,\mathsf{\Pi }\odot V=0,\hfill \end{array}$$

where ⊙ denotes the component-wise multiplication.

Theorem 71 

(Theorem 4.1 and Corollary 4.1, [279]). Let the sequence $\{W_k=(X_k,Y_k,U_k,V_k,{\mathsf{\Lambda }}_k,$${\mathsf{\Pi }}_k\left)\right\}$ be generated by Algorithm 4.

(1) 

Suppose that $\left\{W_k\right\}$ satisfies

$$\begin{array}{c}\hfill \underset{k\to \infty }{lim}(W_{k+1}-W_k)=0.\end{array}$$

Then, any accumulation point $(X,Y,U,V,\mathsf{\Lambda },\mathsf{\Pi })$ of $\left\{W_k\right\}$ satisfies the KKT conditions for Problem (107).

(2) 

If $\left\{W_k\right\}$ converges, then it converges to a KKT point of Problem (107).

Algorithm 3 Extended CGLSA [276] for the real symmetric solution of Equation (5)

Require:  Given matrices: $A,B,C\in {\mathbb{R}}^{n\times m}$; Initial matrices: $X^{\left(0\right)}\in {\mathbb{SR}}^{m\times m}$ and $Y^{\left(0\right)}\in {\mathbb{SR}}^{n\times n}$; Tolerance $ϵ>0$ (for termination criterion); Ensure:  The symmetric solution: $X^{\left(k\right)}$ and $Y^{\left(k\right)}$; 1: The initial residual matrix: $R_0=C-A{X}^{\left(0\right)}-Y^{\left(0\right)}B$; 2: $P_{0,1}={\displaystyle \frac{1}{2}}(A^T{R}_0+R_0^{T}A)$ and $P_{0,2}={\displaystyle \frac{1}{2}}(R_0{B}^T+B{R}_0^T)$; 3: $Q_{0,1}=P_{0,1}$ and $Q_{0,2}=P_{0,2}$; 4: $M_0=A{Q}_{0,1}+Q_{0,2}B$; 5: Set iteration index: $k=0$; 6: while  $\parallel P_{k,1}{\parallel }_F^2+{\parallel P_{k,2}\parallel }_F^2>ϵ$  do 7:        Update iteration index: $k\leftarrow k+1$ 8:        $M_k=A{Q}_{k,1}+Q_{k,2}B$ 9:        ${\alpha }_k={\displaystyle \frac{\parallel P_{k,1}{\parallel }_F^2+{\parallel P_{k,2}\parallel }_F^2}{\parallel M_k{\parallel }_F^2}}$ 10:       Update solution matrices: $X^{(k+1)}=X^{\left(k\right)}+{\alpha }_k{Q}_{k,1}$ and $Y^{(k+1)}=Y^{\left(k\right)}+{\alpha }_k{Q}_{k,2}$; 11:       $R_{k+1}=R_k-{\alpha }_k{M}_k$; 12:       $P_{k+1,1}=P_{k,1}-{\displaystyle \frac{{\alpha }_k}{2}}(A^T{M}_k+M_k^{T}A)$ and $P_{k+1,2}=P_{k,2}-{\displaystyle \frac{{\alpha }_k}{2}}(M_k{B}^T+B{M}_k^T)$; 13:       ${\beta }_k={\displaystyle \frac{\parallel P_{k+1,1}{\parallel }_F^2+{\parallel P_{k+1,2}\parallel }_F^2}{\parallel P_{k,1}{\parallel }_F^2+{\parallel P_{k,2}\parallel }_F^2}}$; 14:       $Q_{k+1,1}=P_{k+1,1}+{\beta }_k{Q}_{k,1}$ and $Q_{k+1,2}=P_{k+1,2}+{\beta }_k{Q}_{k,2}$; 15: end while 16: Output the final solution: $X^{(k+1)}$ and $Y^{(k+1)}$.

Algorithm 4 ADM [279] for the nonnegative solution of Equation (5) over $\mathbb{R}$

Require:  Given matrices: $A\in {\mathbb{R}}^{m\times r}$, $B\in {\mathbb{R}}^{s\times n}$, and $C\in {\mathbb{R}}^{m\times n}$; Initial matrices: $Y_0,U_0,V_0,{\mathsf{\Lambda }}_0,{\mathsf{\Pi }}_0=0$; Penalty parameters: $\alpha ,\beta >0$; $\gamma \in (0,1.618)$; Tolerance $ϵ>0$; Ensure:  The nonnegative solution: $X^{(k+1)}$ and $Y^{(k+1)}$; 1: Set iteration index: $k=0$; 2: while  $k<\mathrm{maxiter}$  do 3:       $X_{k+1}={(A^{T}A+\alpha I)}^{-1}(A^{T}C+\alpha U_k-A^T{Y}_{k}B-{\mathsf{\Lambda }}_k)$; 4:       $Y_{k+1}=(C{B}^T+\beta V_k-A{X}_{k+1}{B}^T-{\mathsf{\Pi }}_k){(B{B}^T+\beta I)}^{-1}$; 5:       $U_{k+1}={\mathcal{P}}_{+}(X_{k+1}+{\displaystyle \frac{1}{\alpha }}{\mathsf{\Lambda }}_k)$; 6:       $V_{k+1}={\mathcal{P}}_{+}(Y_{k+1}+{\displaystyle \frac{1}{\beta }}{\mathsf{\Pi }}_k)$; 7:       ${\mathsf{\Lambda }}_{k+1}={\mathsf{\Lambda }}_k+\gamma \alpha (X_{k+1}-U_{k+1})$; 8:       ${\mathsf{\Pi }}_{k+1}={\mathsf{\Pi }}_k+\gamma \beta (Y_{k+1}-V_{k+1})$; 9:       if $\parallel A{X}_{k+1}+Y_{k+1}{B-C\parallel }_F\le ϵ$ and $X_{k+1},Y_{k+1}\ge 0$ then 10:           Output $X_{k+1},Y_{k+1}$ and return 11:       end if 12:       $k\leftarrow k+1$; 13: end while

We observe that GSE has numerous generalizations, as discussed in Section 6. Consequently, the iterative algorithms for these generalized forms specialize to the corresponding results of GSE as special cases. Next, for convenience, we present the key algorithms related to these generalizations in an enumerated form.

(I)

In 2006, Peng et al. [280] showed an efficient iterative algorithm for solving the following matrix equation over $\mathbb{R}$:

$$AXB+CYD=E,$$

(108)

where X and Y are unknown. They noted that the algorithm can also be used to construct symmetric, antisymmetric, and bisymmetric solutions of (108) with only minor changes.

(II)

The condition number is an important topic in numerical analysis, characterizing the worst-case sensitivity of problems to input data perturbations. A large condition number indicates an ill-posed problem. Consider the following matrix equation:

$$\left\{\begin{array}{c}AX-YB=C,\hfill \\ DX-YE=F,\hfill \end{array}\right.$$

(109)

where X and Y are unknown.

(i)

In 1996, Kågström and Poromaa [281] presented LAPACK-style algorithms and software for solving Equation (109) over $\mathbb{C}$.

(ii)

In 2007, Lin and Wei [282] studied the perturbation analysis for Equation (109) over $\mathbb{R}$, explicitly deriving the expressions and upper bounds for normwise, mixed, and componentwise condition numbers.

(iii)

In 2013, Diao et al. [283] developed the small sample statistical condition estimation algorithm to evaluate the normwise, mixed, and componentwise condition numbers of Equation (109) over $\mathbb{R}$. In [283], they also investigated the effective condition number for Equation (109) and derived sharp perturbation bounds using this condition number.

(III)

In 2008, Dehghan and Hajarian [284] proposed an iterative algorithm to solve the reflexive solutions of Equation (109) over $\mathbb{R}$.

(IV)

In 2010, Dehghan and Hajarian [285] presented an iterative algorithm for solving the generalized bisymmetric solutions of the generalized coupled Sylvester matrix equation over $\mathbb{R}$:

$$\left\{\begin{array}{c}AXB+CYD=M,\hfill \\ EXF+GYH=N,\hfill \end{array}\right.$$

(110)

where X and Y are unknown generalized bisymmetric matrices.

(V)

In 2012, inspired by the least-squares QR-factorization algorithm in [286], Li and Huang [287] proposed an iterative method to find the best approximate solution of Equation (110) over $\mathbb{R}$, where unknown matrices X and Y are constrained to be symmetric, generalized bisymmetric, or $(R,S)$-symmetric.

(VI)

In 2018, inspired by [288,289], Lv and Ma (Section 3, [290]) proposed a parametric iterative algorithm for Equation (109) over $\mathbb{R}$. Moreover, in (Section 4, [290]), they developed an accelerated iterative algorithm based on this parametric approach. Note that Ref. [289] is a monograph on iterative algorithms for the constrained solutions of matrix equations.

(VII)

Interestingly, in 2024, Ma et al. [291] proposed a Newton-type splitting iterative method for the coupled Sylvester-like absolute value equation $\mathbb{R}$:

$$\left\{\begin{array}{c}{A}_{1}X{B}_1+C_1\left|Y\right|D_1=E_1,\hfill \\ A_{2}Y{B}_2+C_2\left|X\right|D_2=E_2,\hfill \end{array}\right.$$

where X and Y are unknown. Here, $\left|A\right|$ means that each component of a matrix A is absolute-valued.

(VIII)

The algorithms in [292,293,294] suffer from parameter tuning challenges, particularly for large-scale problems. To address this limitation, in 2025, Shirilord and Dehghan [295] recently proposed an advanced gradient descent-based parameter-free method to solve Equation (109) over $\mathbb{R}$.

In the final part of this section, we further enumerate some iterative algorithms for solving the generalizations of GSE, which involve an arbitrary number of unknown (or coefficient) matrices.

(A)

In 2005–2006, using the hierarchical identification principle, Ding and Chen [293,294] presented a large family of iterative methods for the more general form of Equation (5) over $\mathbb{R}$, i.e.,

$$A_{i,1}{X}_1{B}_{i,1}+A_{i,2}{X}_2{B}_{i,2}+\dots +A_{i,p}{X}_p{B}_{i,p}=C_i,i=1,2,\dots ,p,$$

(111)

where $X_1,X_2,\dots ,X_p$ are unknown. These iterative methods subsume the well-known Jacobi and Gauss–Seidel iterations. Subsequent scholars have conducted more extensive research on numerical algorithms for Equation (111).

(a)

In 2010, based on the conjugate gradient method, Dehghan and Hajarian [296] constructed an iterative method for Equation (111) over $\mathbb{R}$ with the generalized bisymmetric solutions $(X_1,X_2,\dots ,X_p)$.

(b)

In 2012, Dehghan and Hajarian [297] introduced two iterative methods for solving (111) over $\mathbb{R}$ with the generalized centro-symmetric and central antisymmetric solutions $(X_1,X_2,\dots ,X_p)$.

(c)

In 2014, Hajarian [298] solved Equation (111) over $\mathbb{C}$ by the matrix form of the conjugate gradients squared method.

(d)

In 2016, Hajarian [299] presented the generalized conjugate direction algorithm for computing Equation (111) with the symmetric solutions $(X_1,X_2,\dots ,X_p)$.

(e)

In 2017, based on the Hestenes–Stiefel version of the biconjugate residual (BCR) algorithm, Hajarian [300] solved the generalized Sylvester matrix equation

$$\sum _{i=1}^f\left(A_{i}X{B}_i\right)+\sum _{j=1}^g\left(C_{j}Y{D}_j\right)=E$$

over $\mathbb{R}$ with the generalized reflexive solutions $(X,Y)$. In 2018, Lv and Ma [301] introduced another Hestenes–Stiefel version of BCR method for computing the centrosymmetric or anti-centrosymmetric solutions of Equation (111) over $\mathbb{R}$.

(f)

In 2018, inspired by [302], Sheng [292] proposed a relaxed gradient-based iterative (abbreviated as RGI) algorithm to solve Equation (109), and further generalized this algorithm to Equation (111). Moreover, numerical examples in [292] demonstrate that the RGI algorithm outperforms the iterative algorithm in [294] in terms of speed, elapsed time, and iterative steps.

(g))

In 2018, Hajarian [303] extended the Lanczos version of BCR algorithm to find the symmetric solutions $(X,Y,Z)$ of the matrix equation over $\mathbb{R}$:

$$A_{i}X{B}_i+C_{i}Y{D}_i+E_{i}Z{F}_i=G_i,i=1,2,\dots ,t.$$

In 2020, Yan and Ma [304] also used the Lanczos version of BCR algorithm to study Equation (111) over $\mathbb{R}$ with the (anti-)reflexive solutions $(X_1,X_2,\dots ,X_p)$.

(B)

In 2009, from an optimization perspective, Zhou et al. [305] developed a novel iterative method for solving Equation (111) over $\mathbb{R}$ and its more general form, i.e.,

$$\sum _{j=1}^{s_{i,1}}{A}_{i,1,j}{X}_1{B}_{i,1,j}+\sum _{j=1}^{s_{i,2}}{A}_{i,2,j}{X}_2{B}_{i,2,j}+\dots +\sum _{j=1}^{s_{i,p}}{A}_{i,p,j}{X}_p{B}_{i,p,j}=C_i,i=1,2,\dots ,p$$

with unknown $X_1,X_2,\dots ,X_p$, which contains iterative methods in [293,294] as special cases. In 2015, by extending the generalized product biconjugate gradient algorithms, Hajarian gave [306] four effective matrix algorithms for the coupled matrix equation over $\mathbb{R}$:

$$\sum _{j=1}^l\left(A_{i,1,j}{X}_1{B}_{i,1,j}+A_{i,2,j}{X}_2{B}_{i,2,j}+\dots +A_{i,l,j}{X}_l{B}_{i,l,j}\right)=D_i,i=1,2,\dots ,l,$$

where $X_1,X_2,\dots ,X_l$ are unknown.

(C)

In 2011, Wu et al. [307] constructed an iterative algorithm to solve the coupled Sylvester-conjugate matrix equation over $\mathbb{C}$:

$$\sum _{j=1}^p\left(A_{ij}{X}_j{B}_{ij}+C_{ij}{\overline{X}}_j{D}_{ij}\right)=F_i,i=1,2,\dots ,n,$$

where $X_1,X_2,\dots ,X_p$ are unknown. In 2021, inspired by [307], Yan and Ma [308] proposed an iterative algorithm for the generalized Hamiltonian solutions of the generalized coupled Sylvester-conjugate matrix equations over $\mathbb{C}$:

$$\left\{\begin{array}{c}{\displaystyle \sum _{j=1}^q\left(A_{ij}{X}_j{B}_{ij}+C_{ij}{\overline{Y}}_j{D}_{ij}\right)=M_i,}\hfill \\ {\displaystyle \sum _{j=1}^q\left(E_{ij}{Y}_j{F}_{ij}+G_{ij}{\overline{X}}_j{H}_{ij}\right)=N_i,}\hfill \end{array}\right.$$

where $i=1,\dots ,p$, and $X_j$ and $Y_j$ ($j=1,\dots ,q$) are unknown generalized Hamiltonian matrices.

(D)

In 2015, inspired by [309,310], Hajarian [311] obtained an iterative method for the coupled Sylvester-transpose matrix equations over $\mathbb{R}$:

$$\left\{\begin{array}{c}{\displaystyle \sum _{k=1}^l\left(A_{1,k}X{B}_{1,k}+C_{1,k}{X}^T{D}_{1,k}+E_{1,k}Y{F}_{1,k}\right)=M_1,}\hfill \\ {\displaystyle \sum _{k=1}^l\left(A_{2,k}X{B}_{2,k}+C_{2,k}{X}^T{D}_{2,k}+E_{2,k}Y{F}_{2,k}\right)=M_2}\hfill \end{array}\right.$$

with unknown X and Y, by developing the biconjugate A-orthogonal residual and the conjugate A-orthogonal residual squared methods. Based on this developed method, Hajarian [311] also considered the coupled periodic Sylvester matrix equations over $\mathbb{R}$:

$$\left\{\begin{array}{c}{A}_{1,j}{X}_j{B}_{1,j}+C_{1,j}{X}_{j+1}{D}_{1,j}+E_{1,j}{Y}_j{F}_{1,j}=M_{1,j},\hfill \\ A_{2,j}{X}_j{B}_{2,j}+C_{2,j}{X}_{j+1}{D}_{2,j}+E_{2,j}{Y}_j{F}_{2,j}=M_{2,j},\hfill \end{array}\right.\mathrm{for}j=1,2,\dots ,$$

where $X_j$ and $Y_j$ are unknown periodic matrices with a period.

(E)

Discrete-time periodic matrix equations are an important tool for analyzing and designing periodic systems [312]. More related studies are as follows:

(a)

In 2017, Hajarian [313] introduced a generalized conjugate direction method for solving the general coupled Sylvester discrete-time periodic matrix equations over $\mathbb{R}$:

$$\left\{\begin{array}{cc}{\displaystyle \sum _{j=1}^m\left(A_{ij}{X}_i{B}_{ij}+C_{ij}{X}_{i+1}{D}_{ij}+E_{ij}{Y}_i{F}_{ij}+G_{ij}{Y}_{i+1}{H}_{ij}\right)=M_i,}\hfill & i=1,2,\dots ,\hfill \\ {\displaystyle \sum _{j=1}^m\left({\widehat{A}}_{ij}{X}_i{\widehat{B}}_{ij}+{\widehat{C}}_{ij}{X}_{i+1}{\widehat{D}}_{ij}+{\widehat{E}}_{ij}{Y}_i{\widehat{F}}_{ij}+{\widehat{G}}_{ij}{Y}_{i+1}{\widehat{H}}_{ij}\right)={\widehat{M}}_i,}\hfill & i=1,2,\dots ,\hfill \end{array}\right.$$

where $X_i$ and $Y_i$ are unknown periodic matrices with a period.

(b)

In 2022, Ma and Yan [314] proposed a modified conjugate gradient algorithm for solving the general discrete-time periodic Sylvester matrix equations over $\mathbb{R}$:

$$\sum _{j=1}^h\left(A_{ij}{X}_i{B}_{ij}+C_{ij}{X}_{i+1}{D}_{ij}+E_{ij}{Y}_i{F}_{ij}+G_{ij}{Y}_{i+1}{H}_{ij}\right)=M_i,i=1,2,\dots ,T,$$

where $X_i$ and $Y_i$ are unknown periodic matrices of period T.

(F)

Interestingly, in 2014, Dehghani-Madiseh and Dehghan [315] presented the generalized interval Gauss–Seidel iteration method for the outer estimation of the AE-solution set of the interval generalized Sylvester matrix equation over $\mathbb{R}$:

$$\sum _{i=1}^p{A}_i{X}_i+\sum _{j=1}^q{Y}_j{B}_j=C,$$

where $X_i$ ($i=1,\dots ,p$) and $Y_j$ ($j=1,\dots ,q$) are unknown interval matrices.

(G)

In 2018, Hajarian [316] established the biconjugate residual algorithm for solving the matrix equation over $\mathbb{R}$:

$$\sum _{i=1}^s{A}_{i}X{B}_i+\sum _{j=1}^t{C}_{j}Y{D}_j=M,$$

where X and Y are the unknown generalized reflexive and anti-reflexive matrices, respectively.

(H)

In 2022, based on Kronecker product approximations, Li et al. [317] established a preconditioned modified conjugate residual method for solving the following tensor equation over $\mathbb{R}$:

$$\left\{\begin{array}{c}{\mathcal{X}}_1{\times }_1{A}_{11}+{\mathcal{X}}_2{\times }_2{A}_{12}+\dots +{\mathcal{X}}_{n-1}{\times }_{n-1}{A}_{1(n-1)}+{\mathcal{X}}_n{\times }_n{A}_{1n}={\mathcal{B}}_1,\hfill \\ {\mathcal{X}}_2{\times }_1{A}_{21}+{\mathcal{X}}_3{\times }_2{A}_{22}+\dots +{\mathcal{X}}_n{\times }_{n-1}{A}_{2(n-1)}+{\mathcal{X}}_1{\times }_n{A}_{2n}={\mathcal{B}}_2,\hfill \\ ⋮⋮⋮\hfill \\ {\mathcal{X}}_n{\times }_1{A}_{n1}+{\mathcal{X}}_1{\times }_2{A}_{n2}+\dots +{\mathcal{X}}_{n-2}{\times }_{n-1}{A}_{n(n-1)}+{\mathcal{X}}_{n-1}{\times }_n{A}_{nn}={\mathcal{B}}_n,\hfill \end{array}\right.$$

where ${\mathcal{X}}_1,\dots ,{\mathcal{X}}_n$ are unknown. Here, $\mathcal{A}{\times }_k\mathcal{B}$ denotes the k-mode product [182] of a tensor $\mathcal{A}\in {\mathbb{R}}^{I_1\times I_2\times \dots \times I_n}$ and a matrix $B\in {\mathbb{R}}^{m\times I_k}$.

Remark 69. 

We believe that the iterative algorithms for numerical solutions of GSE and its generalizations summarized in this section can also provide certain inspiration and guidance for the study of numerical solutions to other linear equations.

8. Applications to GSE

In this section, we mainly introduce several applications of GSE in both theoretical and practical problems. It is also worth noting that the generalizations of GSE find applications in pole and eigenstructure assignment [318], scalar functional observer design [319], robots and acoustic source localization [320], pseudo-differential system [321], control theory [249], parametric control [322], model reference tracking control [323], etc.

8.1. Theoretical Applications

8.1.1. Solvability of Matrix Equations

In 1988, Wimmer [223] utilized the solvability condition of the polynomial matrix form of Equation (1) (i.e., Theorem 55) to give a necessary and sufficient condition for the consistency of the matrix equation over a field $\mathbb{F}$:

$$X-AXB=C,$$

(112)

where $A\in {\mathbb{F}}^{p\times p}$, $B\in {\mathbb{F}}^{q\times q}$, and $C\in {\mathbb{F}}^{p\times q}$.

Theorem 72 

(Theorem 2, [223]). Equation (112) has a solution $X\in {\mathbb{F}}^{p\times q}$ if and only if there exist nonsingular matrices $S,R\in {\mathbb{F}}^{(p+q)\times (p+q)}$ such that

$$S\left(\lambda \left[\begin{array}{cc}I& 0\\ 0& B\end{array}\right]+\left[\begin{array}{cc}A& C\\ 0& I\end{array}\right]\right)R=\left(\lambda \left[\begin{array}{cc}I& 0\\ 0& B\end{array}\right]+\left[\begin{array}{cc}A& 0\\ 0& I\end{array}\right]\right).$$

Remark 70. 

The core of the proof of Theorem 72 is that

$$\left(112\right)\iff \left\{\begin{array}{c}X-AY=C,\hfill \\ Y=XB,\hfill \end{array}\right.\iff (A+\lambda I)Y-X(I+\lambda B)=-C.$$

Furthermore, using a special polynomial matrix form of Equation (5), Huang and Liu [237] considered the solvability condition for the matrix equation

$$\sum _{i=0}^k{A}^{i}X{B}_i=C$$

(113)

with unknown X over a ring with identity.

Theorem 73 

(Theorems 1 and 2, [237]). Let $\mathbb{F}$ be a ring with identity, and let $A\in {\mathbb{F}}^{n\times n}$, $B_i\in {\mathbb{F}}^{m\times q}(i=0,1,\dots ,k)$, and $C\in {\mathbb{F}}^{n\times q}$. Denote

$$B\left(\lambda \right)=\sum _{i=0}^k{B}_i{\lambda }^i\in {\mathbb{F}}^{m\times q}\left[\lambda \right].$$

(1) 

Then, Equation (113) is solvable if and only if the polynomial matrix equation

$$(\lambda I-A)X\left(\lambda \right)+Y\left(\lambda \right)B\left(\lambda \right)=C$$

has a solution pair $X\left(\lambda \right)\in {\mathbb{F}}^{n\times q}\left[\lambda \right]$ and $Y\left(\lambda \right)\in {\mathbb{F}}^{n\times m}\left[\lambda \right]$.

(2) 

Suppose that $\mathbb{F}$ is a division ring and A is algebraic (or $\mathbb{F}$ is a finitely generated as module over its center). Then, Equation (113) is solvable if and only if

$$\left[\begin{array}{cc}\lambda I-A& -C\\ 0& B\left(\lambda \right)\end{array}\right]\mathit{and}\left[\begin{array}{cc}\lambda I-A& 0\\ O& B\left(\lambda \right)\end{array}\right]$$

are equivalent.

Remark 71. 

(1) 

Theorem 73 remains valid when $\mathbb{F}$ is a finite dimensional central simple algebra over a field (see Theorems 3 and 4, [235]).

(2) 

In Corollaries 1–3 of [237], Huang and Liu indicated that relevant results regarding the solvability of the equations $AX-XB=C$, $X-AXB=C$, and $AXB=C$ can be directly derived by Theorem 73.

8.1.2. UTV Decomposition of Dual Matrices

In 2024, Xu et al. [324] presented the UTV decomposition of dual complex matrices based on the solvability conditions and general solution representations of Equation (1) over $\mathbb{C}$ (i.e., Theorem 3).

For $a_s,a_i\in \mathbb{C}$, $a=a_s+a_i\epsilon$ represents a dual complex number, where $\epsilon$ is the dual unit given in (60). The set of all dual complex numbers is denoted by $\mathbb{DC}$. For $A=A_s+A_i\epsilon \in {\mathbb{DC}}^{n\times p}$, A has unitary columns if $n\ge p$ and $A^{*}A=I_n$, where $A^{*}=A_s^{*}+A_i^{*}\epsilon$ is the conjugate transpose of A. For $A=A_s+A_i\epsilon \in {\mathbb{DC}}^{n\times n}$, A is unitary if $A^{*}A=A{A}^{*}=I_n$; A is diagonal if both $A_s$ and $A_i$ are diagonal; and A is nonsingular if $A{A}^{-1}=A^{-1}A=I_n$ for some $A^{-1}\in {\mathbb{DC}}^{n\times n}$.

We say that $A=A_s+A_i\epsilon \in {DC}^{m\times n}$ has the UTV decomposition [324] if

$$A=UT{V}^{*},$$

(114)

where $U=U_s+U_i\epsilon \in {\mathbb{DC}}^{m\times k}$ has unitary columns, $T=T_s\in {\mathbb{C}}^{k\times k}$ is triangular and nonsingular, and $V=V_s+V_i\epsilon \in {\mathbb{DC}}^{n\times k}$ has a unitary standard part $V_s$.

Theorem 74 

(Theorem 3.1, [324]). Let $A=A_s+A_i\epsilon \in {DC}^{m\times n}$. Assume that the UTV decomposition of $A_s$ is given by

$$A_s=U_s{T}_s{V}_s^{*},$$

where both $U_s\in {\mathbb{C}}^{m\times k}$ and $V_s\in {\mathbb{C}}^{n\times k}$ have unitary columns, and $T_s\in {\mathbb{C}}^{k\times k}$ is triangular and nonsingular. Then, the UTV decomposition of A exists if and only if

$$(I_m-U_s{U}_s^{*})A_i(I_n-V_s{V}_s^{*})=0,$$

in which case,

$$U_i=(I_m-U_s{U}_s^{*})A_i{V}_s{T}_s^{-1}+U_{s}P\mathit{and}{V}_i=A_i^{*}{U}_s{\left(T_s^{-1}\right)}^{*}-V_s{T}_s^{*}{P}^{*}{\left(T_s^{-1}\right)}^{*},$$

where $P\in {\mathbb{C}}^{k\times k}$ is arbitrary skew-Hermitian matrix.

Remark 72. 

The proof of Theorem 3.1 in [324] shows that the pivotal step in proving Theorem 74 is that A has UTV decomposition if and only if the matrix equation

$$U_i{T}_s{V}_s^{*}+U_s{T}_s{V}_i^{*}=A_i$$

is consistent for unknown $U_i\in {\mathbb{DC}}^{m\times k}$ and $V_i\in {\mathbb{DC}}^{n\times k}$.

8.1.3. Microlocal Triangularization of Pseudo-Differential Systems

In 2013, using the solvability of Equation (1) over $\mathbb{C}$, Kiran [321] constructed a recursive scheme to factorize a pseudo-differential system into lower and upper triangular systems (LU factorization) independent of lower order terms.

Let $OP{S}_N^0(\Omega )$ be the set of all $N\times N$ pseudo-differential systems of order 0 defined on an open subset $\Omega$ of ${\mathbb{R}}^n$. In addition, the notation and terminology in this subsection follow that in [325,326].

Definition 20 

(Definition 2.2, [321]). A matrix valued operator $A\in OP{S}_N^0(\Omega )$ admits $LU$ factorization if

$$A=LU,$$

where $L\in OP{S}_N^0(\Omega )$ is an elliptic lower triangular matrix whose principal symbol has the identity on the diagonal entries, and $U\in OP{S}_N^0(\Omega )$ is an upper triangular matrix.

Theorem 75 

(Theorem 2.3, [321]). Let ${\lambda }_1(x,\xi ),\dots ,{\lambda }_N(x,\xi )$ be N sections of eigenvalues of the principal symbol of $A\in OP{S}_N^0(\Omega )$, including multiplicities, in a conic neighborhood Γ of $(x_0,{\xi }_0)\in T^{*}\Omega \setminus \left\{0\right\}$. If

$${\lambda }_i^{-1}\left(0\right)\cap {\lambda }_j^{-1}\left(0\right)\cap \mathsf{\Gamma }=\varnothing ,i\ne j,$$

then A is microlocally triangularizable in Γ independent of lower order terms. Moreover, the system admits $LU$ factorization independent of lower order terms in Γ if and only if the principal symbol of A admits an $LU$ factorization where the first $N-1$ eigenvalues of the upper triangular matrix do not vanish in Γ.

Remark 73. 

(1) 

In Sections 3.3 and 3.4 of [321], Kiran showed that the triangularization scheme in Theorem 75 can also be applied to symbolic hierarchies.

(2) 

Lemma 2.5 of [321] shows that Equation (1) over $\mathbb{C}$ has a unique solution X if and only if A or B is nonsingular. However, there is a simple counterexample to its sufficiency. Indeed, if both A and B are identity matrices (and thus nonsingular), the solution X of Equation (1) is obviously not unique for a given C. For instance, take $X=C$ and $Y=0$, or $X=2C$ and $Y=C$. This minor error, however, does not affect the existence of solutions to Equation (1).

8.2. Practical Applications

8.2.1. Calibration Problems

The late 1970s to early 1980s witnessed a surge of interest in deploying robotic manipulators for automated manufacturing. However, integrating robots as core components in flexible manufacturing systems remained challenging across many industrial applications, prompting extensive research into the manipulator calibration [327].

Inspired by [328,329], Zhuang et al. [330] first solved a specialized form of Equation (1) to address a robot calibration problem: the calibration of the robot/world (i.e., the BASE transformation) and tool/flange (i.e., the TOOL transformation). As a matter of fact, robot manipulator calibration refers to the procedure of enhancing a robot manipulator’s accuracy by adjusting its control software.

Figure 1 provides a schematic illustration of the geometry of a robotic cell. The world coordinate frame serves as an external reference frame. The base coordinate frame is defined within the robot structure. The flange coordinate frame is defined on the mounting surface of the robot end effector. The tool frame is positioned at a point inside the end effector.

Then, the robot kinematic model can be transformed into the following form:

$$\mathbf{A}\mathbf{X}=\mathbf{Y}\mathbf{B},$$

(115)

where

(i)

$\mathbf{A}$ is the known homogeneous transformation from end effector pose measurements,

(ii)

$\mathbf{B}$ is derived from the calibrated manipulator internal-link forward kinematics;

(iii)

$\mathbf{X}$ is the unknown transformation from the tool frame to the flange frame;

(iv)

$\mathbf{Y}$ is the unknown transformation from the world frame to the base frame.

Assume that there are n pose measurements for $i=1,2,\dots n$. Thus, the calibration problem is reduced to solving the system of equations

$${\mathbf{A}}_i\mathbf{X}={\mathbf{Y}}_i\mathbf{B},i=1,2,\dots n.$$

Subsequently, Zhuang et al. elaborated on the solution of Equation (115) using the rotational properties of quaternions in Section 3 of [330].

Subsequently, the robot manipulator calibration problem is further optimized by considering Equation (115) through different approaches: dual quaternion method [331,332,333,334,335], new hybrid calibration method [336], least-squares approach [337], Kronecker product method [338], 3D position measurements [339], nonlinear optimization and evolutionary computation [340], 2D positional features [341], dual Lie algebra [342], symbolic method [343], linear matrix inequality and semi-definite programming optimization [344], probabilistic framework [345], transference principle [346], etc.

8.2.2. Encryption and Decryption Schemes for Color Images

The RGB (red, green, blue) color channels can be directly mapped to the imaginary parts ($\mathbf{i}$, $\mathbf{j}$, $\mathbf{k}$) of a pure imaginary quaternion matrix. Naturally, a dual quaternion matrix can represent two color images since both its standard and infinitesimal parts are quaternion matrices. By solving Equation (1) over $\mathbb{DH}$ (i.e., Theorem 43), Xie et al. (Section 4, [159]) proposed the encryption and decryption schemes for color images, as shown in Figure 2.

The specific encryption and decryption processes for color images are presented in Algorithms 5 and 6, respectively. To ensure the uniqueness of the decrypted color images $({\widehat{X}}_0,{\widehat{X}}_1)$ output by Algorithmic 6, the encryption dual quaternion matrices A and B must be restricted to satisfy Condition P, i.e., both the standard parts and the infinitesimal parts of A and B are of either full row rank or full column rank.

Algorithm 5 Color image encryption scheme

1: Input: two original color images $X_0$ and $X_1$, two color images $Y_0$ and $Y_1$ as keys, and two encryption dual quaternion matrices A and B satisfying Condition P; 2: Output: two encrypted color images $C_0$ and $C_1$ by $$A(X_0+X_1\epsilon )-(Y_0+Y_1\epsilon )B=C_0+C_1\epsilon .$$

Algorithm 6 Color image decryption scheme

1: Input: the encryption matrices A and B, the keys $Y_0$ and $Y_1$, and the encrypted color images $C_0$ and $C_1$ from Algorithm 5. 2: Output: two decrypted color images ${\widehat{X}}_0$ and ${\widehat{X}}_1$ by (61) and (62) in Theorem 43.

As illustrated in Figure 3, Xie et al. [159] selected two color images (i.e., Bike 1 and Bike 2) as the objects to be encrypted, and another two color images (i.e., Sunflower and Big-Windmill) as the keys. They first encrypted Bike 1 and Bike 2 using Algorithm 5, obtaining the two encrypted color images (see Figure 4). Subsequently, Algorithm 6 was applied to decrypt the encrypted images, resulting in the two decrypted color images (see Figure 5). Moreover, the structural similarity index measures (SSIM) of both decrypted images are shown in

Table 2. SSIM values for the decrypted color images.

Color Image	SSIM
Bike 1	0.99
Bike 2	0.99

, which fully demonstrates the high effectiveness of their decryption scheme.

Remark 74. 

Recently, research on encrypting and decrypting color images and color videos by solving matrix or tensor equations has attracted attention in [203,204,347].

9. Conclusions

Research on GSE and its generalizations has long been a vibrant field, boasting both profound theoretical value and extensive practical application prospects. This comprehensive review encompasses 75 theorems, 74 remarks, and 347 references spanning from 1844 to 2025, covering pure mathematics (linear algebra, abstract algebra, operators, tensors, semi-tensor products, polynomial matrices, etc.), computational mathematics (iterative algorithms, condition numbers, etc.), and applied mathematics (robotics, image processing, encryption/decryption schemes, etc.). Centered on solving GSE, this paper elaborates on five dimensions—methods, constraints, generalizations, algorithms, and applications—distilling the field’s essence through point-by-point analysis and synthesis. A network diagram (i.e., Figure 6) intuitively illustrates the GSE research’s core framework.

As shown in Figure 6, let us start with “$AX-YB=C$ (GSE)” in the red box, which is the core research object of the entire paper. The orange boxes show the various generalizations of GSE in different mathematical fields, which greatly enriches the relevant research. Then, looking down, the red box “SOLUTIONS” represents the main problem discussed in GSE and its generalizations, that is, the problem of solving equations, which is the core of the entire research. The green box on the right imposes the relevant constraint on the solution of the equation due to the demand for special matrices or specific conditions in practical applications. Further down, the purple box is a further elaboration on solving the equation: first, discuss the solvability conditions of the relevant equation; in the solvable case, study the representation of its analytical solution (explicit solution); in the unsolvable case, consider the analytical solution of its best approximation solution (minimum-norm least squares solution); finally, when the relevant analytical solution cannot be obtained, or when constrained by factors such as computational complexity, it is necessary to further study the numerical solution of related problems by designing iterative algorithms and continuously enhance computational efficiency. The blue box on the left contains various methods for tackling related solving problems, and also different approaches that can be chosen to solve the same problem. The gray box at the bottom is the purpose or end-point of all previous theoretical research, that is, to serve applications, which includes two aspects: theoretical applications and practical applications.

In the narrative of this paper, we have provided detailed introductions, in the form of remarks, to several interesting problems worthy of further exploration. For the convenience of readers, we have hereby compiled

Table 3. Open problems for further research on GSE.

Number	Remark Number	Open Problem
1	Remarks 15 and 16	Cramer’s rule for GSE only through coefficient matrices (partially solved)
2	Remark 18	Solving GSE under STP (or MM-2 STP)
3	Remark 22	Solving the GSE via $\mathcal{L}$-representation, $\mathcal{C}$-representation, ${\mathcal{L}}_{\mathcal{C}}$-representation, and vectorization properties of STP, respectively,
4	Remark 35	Discussing RET from a single common property of Euclidean domains and unit regular rings (partially solved)
5	Remark 43	Research on the fundamental properties and applications of combinations of different types of quaternions and dual numbers
6	Remark 47	Investigating GSE tensor equations under different tensor products and quaternion algebras.
7	Remark 60 (4)	Exploring converse problems for Lemmas 11 and 12 of [241]
8	Remark 63	Solving the restricted system (101)
9	Remark 64	Solving the system (104) (partially solved)

to briefly summarize these problems.

Numerous researchers worldwide have made significant contributions to GSE-related studies. Owing to the authors’ limitations, we have not been able to cover all GSE research findings, and there may inevitably be inadequacies. We offer our apologies here. Moreover, as GSE research advances rapidly with continuous innovations, this paper’s framework and content will grow increasingly rich and substantial as developments unfold. Finally, we kindly request experts and readers to offer their valuable insights, assisting us in further refining the summarization work in this field.