The η-Anti-Hermitian Solution to a System of Constrained Matrix Equations over the Generalized Segre Quaternion Algebra

Abstract

In this paper, we propose three real representations of a generalized Segre quaternion matrix. We establish necessary and sufficient conditions for the existence of the $\eta$-anti-Hermitian solution to a system of constrained matrix equations over the generalized Segre quaternion algebra. We also obtain the expression of the general $\eta$-anti-Hermitian solution to the system when it is solvable. Finally, we provide a numerical example to verify the main results of this paper.

1. Introduction

In 1843, Hamilton [1] discovered the real quaternions

$$\mathbb{H}=\{q=q_0+q_{1}i+q_{2}j+q_{3}k:i^2=j^2=k^2=-1,ijk=-1,q_0,q_1,q_2,q_3\in \mathbb{R}\}$$

which is a four-dimensional non-commutative division algebra over the real number field $\mathbb{R}.$

The real quaternions have played an important role in many fields such as quantum physics, computer graphics and signal processing [2,3,4]. In these areas, the real quaternion algebra is more useful than the usual algebra. For example, Ling et al. [5] presented a new algorithm for solving the linear least squares problem over the quaternions. By means of direct quaternion arithmetics, the algorithm does not make the scale of the problem dilate exponentially, compared to the conventional real or complex representation methods. However, the multiplication of real quaternions is non-commutative; therefore, to avoid non-commutativity, the commutative quaternion algebra was introduced.

In 1892, Segre [6] defined the commutative quaternions

$$\mathbb{S}=\left\{a=a_0+a_{1}i+a_{2}j+a_{3}k:i^2=-1,j^2=1,ij=ji=k,a_0,a_1,a_2,q{a}_3\in \mathbb{R}\right\}$$

which is a four-dimensional commutative algebra that is not divisible over $\mathbb{R}.$

The commutative quaternions have been widely applied in various fields. For color image processing, Pei et al. [7] defined a simplified commutative quaternion polar form to represent color images, which is useful in the brightness–hue–saturation color space. After this, Pei et al. [8] developed the algorithms for calculating the eigenvalues, the eigenvectors and the singular value decompositions of commutative quaternion matrices. They employed the singular value decompositions of commutative quaternion matrices to implement a color image which reduces the computational complexity to one-forth of the conventional. Guo et al. [9] defined the reduced canonical transform of commutative quaternions which is the generalization of reduced Fourier transform of commutative quaternions. Lin et al. [10] established a commutative quaternion valued neural network (CQVNN) and studied the asymptotic stability of CQVNN.

The commutative quaternion matrix equations have been studied extensively. In [11], Kosal et al. studied some algebraic properties of commutative quaternion matrices using complex representations. Kosal et al. [12] gave the expression of the general solution to the Kalman–Yakubovich conjugate matrix equation over the commutative quaternions by means of real representations of a commutative quaternion matrix. Moreover, Kosal et al. [13] studied Sylvester-conjugate commutative quaternion matrix equations using real representations. In [14], Kosal et al. proposed a different kind of real representation of commutative quaternion matrices, and they also studied the general solution to matrix equation $AX=B$ over the commutative quaternions.

Segre [6] extended the commutative quaternion algebra to the generalized Segre quaternion algebra ${\mathbb{S}}_g$, which is defined as follows:

$${\mathbb{S}}_g=\{a=a_0+a_{1}i+a_{2}j+a_{3}k:a_0,a_1,a_2,a_3\in \mathbb{R}andi,j,k\notin \mathbb{R}\},$$

where $i,j,k$ satisfy the following multiplication rules:

$$i^2=k^2=\alpha ,j^2=1,ij=ji=k,jk=kj=i,ik=ki=\alpha j.$$

Here, we only consider the case in which $\alpha \ne 0$. In particular, ${\mathbb{S}}_g$ is the commutative quaternions $\mathbb{S}$ when $\alpha =-1$.

The generalized Segre quaternion algebra, which includes the commutative quaternions, shows the superiority of the proposed approach in signal processing over its counterpart in the real quaternions. Moreover, the Segre quaternions have potential applications in linear models, filtering and smoothing as well as signal detection [15].

The commutative quaternion algebra has many vital applications to areas of mathematics and physics. On this basis, the generalized Segre quaternion algebra is rarely studied. Out of this motivation, we focus on the generalized Segre quaternion algebra in this paper.

For $A\in {\mathbb{S}}_g^{m\times n}$, A can be uniquely expressed as $A=A_0+A_{1}i+A_{2}j+A_{3}k$, where $A_0,A_1,A_2,A_3\in {\mathbb{R}}^{m\times n}$. We define $\eta$-conjugates [11], $\eta \in \{i,j,k\}$, as follows:

$$\begin{array}{c}{A}^i=A_0-A_{1}i+A_{2}j-A_{3}k,\hfill \\ A^j=A_0+A_{1}i-A_{2}j-A_{3}k,\hfill \\ A^k=A_0-A_{1}i-A_{2}j+A_{3}k,\hfill \end{array}$$

and $\eta$-conjugate transposes, $\eta \in \{i,j,k\}$, as follows:

$$\begin{array}{c}{A}^{i*}={A_0}^T-A_1^{T}i+A_2^{T}j-A_3^{T}k,\hfill \\ A^{j*}=A_0^T+A_1^{T}i-A_2^{T}j-A_3^{T}k,\hfill \\ A^{k*}=A_0^T-A_1^{T}i-A_2^{T}j+A_3^{T}k.\hfill \end{array}$$

Definition 1. 

For $\eta \in \{i,j,k\}$, $A\in {\mathbb{S}}_g^{n\times n}$ is called η-Hermitian matrix if $A=A^{\eta *}$, $A\in {\mathbb{S}}_g^{n\times n}$ is called η-anti-Hermitian matrix if $A=-A^{\eta *}$.

The concept of $\eta$-(anti)-Hermitian matrix is a more generalized concept comparing with (skew-)Hermitian matrix. $\eta$-(anti)-Hermitian matrix has important applications in linear modeling and convergence analysis in statistical signal processing [16,17,18]. The $\eta$-(anti)-Hermitian solutions to matrix equations over real quaternions have been vastly investigated [19,20,21,22]. However, the $\eta$-anti-Hermitian solutions to matrix equations over the commutative quaternion algebra have received little attention. Yuan et al. [23] investigated the Hermitian solutions of commutative quaternion matrix equation $(AXB,CXD)=(E,G)$, which has wide applications in control and system theory, stability theory and neural network. Tian et al. [24] studied the anti-Hermitian solutions of matrix equations $AX{A}^H+BY{B}^H=C$ over the commutative quaternion algebra.

Motivated by significant application and research value of $\eta$-(anti)-Hermitian matrix and matrix equations, we consider the $\eta$-anti-Hermitian solution to the generalized Segre quaternion matrix equation

$$AX=B$$

(1)

and a system of constrained generalized Segre quaternion matrix equations

$$\begin{array}{cc}\hfill & E_{1}X{E}_2+F_{1}Y{F}_2=H,\hfill \\ \hfill & s.t.\left\{\begin{array}{ccc}\hfill A_{1}X& =\hfill & B_1,\hfill \\ \hfill X{A}_2& =\hfill & B_2,\hfill \\ \hfill C_{1}Y& =\hfill & D_1,\hfill \\ \hfill Y{C}_2& =\hfill & D_2,\hfill \end{array}\right.\hfill \end{array}$$

(2)

where X and Y are unknown matrices and the other matrices are given with appropriate orders. The classic matrix equation $AX=B$ over the real quaternions is important in various applications, for example, Yuan et al. [25] discussed its application in color image restoration. However, to the best of our knowledge, currently known real representations of commutative matrices are not able to address the $\eta$-anti-Hermitian solutions of matrix equations. Our scientific innovation lies in not only establishing three different real representations, making up for the result of $\eta$-anti-Hermitian solutions over the commutative quaternions, but also generalizing it to the Segre quaternion algebra which is more extensive and of application value.

2. Preliminaries

In this section, we specify the notations of the paper and propose three real representations of the matrix over the generalized Segre quaternion algebra. The algebraic properties of the real representations are also given.

2.1. Notations

Throughout this paper, we use the following notations:

• $I_n$ denotes the $n\times n$ identity matrix;
• $A^T$, $rank\left(A\right)$ denote the transpose and rank of a matrix A, respectively;
• $A^{†}$ denotes the Moore–Penrose inverse of A, which satisfies simultaneously $A{A}^{†}A=A$, $A^{†}A{A}^{†}=A^{†}$, ${\left(A{A}^{†}\right)}^{*}=A{A}^{†}$ and ${\left(A^{†}A\right)}^{*}=A^{†}A$. Moreover, $L_A=I-A^{†}A$ and $R_A=I-A{A}^{†}$ are two projectors induced by A, respectively;
• $A\otimes B=\left(a_{pq}B\right)\in {\mathbb{R}}^{mt\times ns}$ denotes the Kronecker product of matrices $A=\left(a_{pq}\right)\in {\mathbb{R}}^{m\times n}$ and $B=\left(b_{pq}\right)\in {\mathbb{R}}^{t\times s}$;
• $Vec\left(A\right)={({x_1}^T,{x_2}^T,\cdots ,{x_n}^T)}^T\in {\mathbb{R}}^{mn}$, where $x_i(i=1,\cdots ,n)$ is the i-th column vector of A, denotes the stretching function of a matrix A.

2.2. Real Representations

Theorem 1. 

Let $A\in {\mathbb{S}}_g^{m\times n}$ and $B\in {\mathbb{S}}_g^{n\times s}$. Then,

(1) 

${\left(A^{\eta }\right)}^T={\left(A^T\right)}^{\eta }=A^{\eta *}$,

(2) 

${\left(AB\right)}^{\eta }=A^{\eta }{B}^{\eta }$,

(3) 

${\left(AB\right)}^T=B^T{A}^T$,

(4) 

${\left(AB\right)}^{\eta *}=B^{\eta *}{A}^{\eta *}$,

(5) 

${\left(A^{\eta *}\right)}^{\eta *}=A$.

Proof. 

For (2), it can be obtained by the proof of Theorem 3.1 in [11]. The others can be verified easily. □

Let $A\in {\mathbb{S}}_g^{m\times n}$, $A=A_0+A_{1}i+A_{2}j+A_{3}k$, where $A_0,A_1,A_2,A_3\in {\mathbb{R}}^{m\times n}$. By generalizing the real representation in [14], we define a kind of real representations of A as follows:

$$A^{{\sigma }_i}=\left(\begin{array}{cccc}{A}_0& \alpha A_1& A_2& \alpha A_3\\ A_1& A_0& A_3& A_2\\ A_2& \alpha A_3& A_0& \alpha A_1\\ A_3& A_2& A_1& A_0\end{array}\right).$$

Similarly, we define two other kinds as follows:

$$A^{{\sigma }_j}=V_m{A}^{{\sigma }_i}=\left(\begin{array}{cccc}-A_0& -\alpha A_1& -A_2& -\alpha A_3\\ A_1& A_0& A_3& A_2\\ A_2& \alpha A_3& A_0& \alpha A_1\\ -A_3& -A_2& -A_1& -A_0\end{array}\right),A^{{\sigma }_k}=W_m{A}^{{\sigma }_i}=\left(\begin{array}{cccc}{A}_0& \alpha A_1& A_2& \alpha A_3\\ A_1& A_0& A_3& A_2\\ -A_2& -\alpha A_3& -A_0& -\alpha A_1\\ -A_3& -A_2& -A_1& -A_0\end{array}\right),$$

where

$$V_m=\left(\begin{array}{cccc}-I_m& 0& 0& 0\\ 0& I_m& 0& 0\\ 0& 0& I_m& 0\\ 0& 0& 0& -I_m\end{array}\right)and{W}_m=\left(\begin{array}{cccc}{I}_m& 0& 0& 0\\ 0& I_m& 0& 0\\ 0& 0& -I_m& 0\\ 0& 0& 0& -I_m\end{array}\right).$$

Let

$$G_n=\left(\begin{array}{cccc}{I}_n& 0& 0& 0\\ 0& -\alpha I_n& 0& 0\\ 0& 0& I_n& 0\\ 0& 0& 0& -\alpha I_n\end{array}\right),R_n=\left(\begin{array}{cccc}0& \alpha I_n& 0& 0\\ I_n& 0& 0& 0\\ 0& 0& 0& \alpha I_n\\ 0& 0& I_n& 0\end{array}\right),$$

$$S_n=\left(\begin{array}{cccc}0& 0& I_n& 0\\ 0& 0& 0& I_n\\ I_n& 0& 0& 0\\ 0& I_n& 0& 0\end{array}\right)and{T}_n=\left(\begin{array}{cccc}0& 0& 0& \alpha I_n\\ 0& 0& I_n& 0\\ 0& \alpha I_n& 0& 0\\ I_n& 0& 0& 0\end{array}\right).$$

By direct calculation, the following properties of three real representations are obtained.

Proposition 1. 

For $A,B\in {\mathbb{S}}_g^{m\times n}$, $C\in {\mathbb{S}}_g^{n\times s}$, $\lambda \in \mathbb{R}$, then

(1) 

$A=B\iff A^{{\sigma }_{\eta }}=B^{{\sigma }_{\eta }},\eta \in \{i,j,k\}$,

(2) 

${(A+B)}^{{\sigma }_{\eta }}=A^{{\sigma }_{\eta }}+B^{{\sigma }_{\eta }},{\left(\lambda A\right)}^{{\sigma }_{\eta }}=\lambda A^{{\sigma }_{\eta }},\eta \in \{i,j,k\},$

(3) 

${\left(AC\right)}^{{\sigma }_i}=A^{{\sigma }_i}{C}^{{\sigma }_i},{\left(AC\right)}^{{\sigma }_j}=A^{{\sigma }_j}{V}_n{C}^{{\sigma }_j},{\left(AC\right)}^{{\sigma }_k}=A^{{\sigma }_k}{W}_n{C}^{{\sigma }_k},$

(4) 

(a) ${R_m}^{-1}{A}^{{\sigma }_i}{R}_n=A^{{\sigma }_i},{S_m}^{-1}{A}^{{\sigma }_i}{S}_n=A^{{\sigma }_i},{T_m}^{-1}{A}^{{\sigma }_i}{T}_n=A^{{\sigma }_i}$,

(b)${R_m}^{-1}{A}^{{\sigma }_j}{R}_n=-A^{{\sigma }_j},{S_m}^{-1}{A}^{{\sigma }_j}{S}_n=-A^{{\sigma }_j},{T_m}^{-1}{A}^{{\sigma }_j}{T}_n=A^{{\sigma }_j},$

(c)${R_m}^{-1}{A}^{{\sigma }_k}{R}_n=A^{{\sigma }_k},{S_m}^{-1}{A}^{{\sigma }_k}{S}_n=-A^{{\sigma }_k},{T_m}^{-1}{A}^{{\sigma }_k}{T}_n=-A^{{\sigma }_k}$

(5) 

${\left(A^{i*}\right)}^{{\sigma }_i}={G_n}^{-1}{\left(A^{{\sigma }_i}\right)}^T{G}_m,{\left(A^{j*}\right)}^{{\sigma }_j}={G_n}^{-1}{\left(A^{{\sigma }_j}\right)}^T{G}_m,{\left(A^{k*}\right)}^{{\sigma }_k}={G_n}^{-1}{\left(A^{{\sigma }_k}\right)}^T{G}_m,$

(6) 

(a)  $A=\frac{1}{4}\left(\begin{array}{cccc}{I}_m& i{I}_m& j{I}_m& k{I}_m\end{array}\right)A^{{\sigma }_i}\left(\begin{array}{c}{I}_n\\ \frac{1}{\alpha }i{I}_n\\ j{I}_n\\ \frac{1}{\alpha }k{I}_n\end{array}\right),$

(b)  $A=\frac{1}{4}\left(\begin{array}{cccc}-I_m& i{I}_m& j{I}_m& -k{I}_m\end{array}\right)A^{{\sigma }_j}\left(\begin{array}{c}{I}_n\\ \frac{1}{\alpha }i{I}_n\\ j{I}_n\\ \frac{1}{\alpha }k{I}_n\end{array}\right),$

(c)  $A=\frac{1}{4}\left(\begin{array}{cccc}{I}_m& i{I}_m& -j{I}_m& -k{I}_m\end{array}\right)A^{{\sigma }_k}\left(\begin{array}{c}{I}_n\\ \frac{1}{\alpha }i{I}_n\\ j{I}_n\\ \frac{1}{\alpha }k{I}_n\end{array}\right).$

3. $\eta$-Anti-Hermitian solution to Equation (1) and the System (2)

In this section, by three real representations and related lemmas, we derive the necessary and sufficient conditions for Equation (1) and the system (2) to have $\eta$-anti-Hermitian solutions over ${\mathbb{S}}_g$ and obtain expressions of the general $\eta$-anti-Hermitian solution in Section 3.1 and Section 3.2, respectively. In Section 3.3, we give a numerical example.

3.1. $\eta$-Anti-Hermitian Solution to Equation (1)

Lemma 1. 

[26] Let $A\in {\mathbb{R}}^{m\times n}$ and $B\in {\mathbb{R}}^{m\times n}$. Then, the matrix equation $AX=B$ has a skew symmetric solution $X=-X^T\in {\mathbb{R}}^{n\times n}$ if and only if

$$R_{A}B=0,B{A}^T=-A{B}^T.$$

In which case, the general skew symmetric solution to $AX=B$ is

$$X=A^{†}B-{\left(A^{†}B\right)}^T+A^{†}A{B}^T{\left(A^{†}\right)}^T+L_{A}U{\left(L_A\right)}^T,$$

where $U=-U^T\in {\mathbb{R}}^{n\times n}$ is an arbitrary matrix.

Theorem 2. 

Let $A\in {{\mathbb{S}}_g}^{m\times n}$, $B\in {{\mathbb{S}}_g}^{m\times n}$, $\eta \in \{i,j,k\},$

$$A_{\eta }=\left\{\begin{array}{cc}{A}^{{\sigma }_i}{G_n}^{-1},& \eta =i\hfill \\ A^{{\sigma }_j}{V}_n{G_n}^{-1},& \eta =j\hfill \\ A^{{\sigma }_k}{W}_n{G_n}^{-1},& \eta =k\hfill \end{array}\right.and{B}_{\eta }=B^{{\sigma }_{\eta }}.$$

Then, the Equation (1) has an η-anti-Hermitian solution $X=-X^{{\eta }^{*}}\in {{\mathbb{S}}_g}^{n\times n}$ if and only if

(1) 

The corresponding real matrix equation $A_{\eta }{Y}_{\eta }=B_{\eta }$ has a skew symmetric solution $Y_{\eta }\in {\mathbb{R}}^{4n\times 4n}$.

(2) 

The following conditions hold:

$$R_{A_{\eta }}{B}_{\eta }=0,B_{\eta }{A_{\eta }}^T=-A_{\eta }{B_{\eta }}^T.$$

(3)

The above two statements are equivalent to each other. In which case, the general η-anti-Hermitian solution to Equation(1)can be expressed as follows:

(a) 

in the case of $\eta =i$,

$$X=\frac{1}{16}\left(\begin{array}{cccc}{I}_n& -\frac{1}{\alpha }i{I}_n& j{I}_n& -\frac{1}{\alpha }k{I}_n\end{array}\right)(Y_i-{R_n}^T{Y}_i{R_n}^{-1}+{S_n}^T{Y}_i{S_n}^{-1}-{T_n}^T{Y}_i{T_n}^{-1})\left(\begin{array}{c}{I}_n\\ \frac{1}{\alpha }i{I}_n\\ j{I}_n\\ \frac{1}{\alpha }k{I}_n\end{array}\right),$$

(b) 

in the case of $\eta =j$,

$$X=\frac{1}{16}\left(\begin{array}{cccc}-I_n& -\frac{1}{\alpha }i{I}_n& j{I}_n& \frac{1}{\alpha }k{I}_n\end{array}\right)(Y_j+{R_n}^T{Y}_j{R_n}^{-1}-{S_n}^T{Y}_j{S_n}^{-1}-{T_n}^T{Y}_j{T_n}^{-1})\left(\begin{array}{c}{I}_n\\ \frac{1}{\alpha }i{I}_n\\ j{I}_n\\ \frac{1}{\alpha }k{I}_n\end{array}\right),$$

(4)

(c) 

in the case of $\eta =k$,

$$X=\frac{1}{16}\left(\begin{array}{cccc}{I}_n& -\frac{1}{\alpha }i{I}_n& -j{I}_n& \frac{1}{\alpha }k{I}_n\end{array}\right)(Y_k-{R_n}^T{Y}_k{R_n}^{-1}-{S_n}^T{Y}_k{S_n}^{-1}+{T_n}^T{Y}_k{T_n}^{-1})\left(\begin{array}{c}{I}_n\\ \frac{1}{\alpha }i{I}_n\\ j{I}_n\\ \frac{1}{\alpha }k{I}_n\end{array}\right).$$

In above (a)–(c),

$$Y_{\eta }={A_{\eta }}^{†}{B}_{\eta }-{\left({A_{\eta }}^{†}{B}_{\eta }\right)}^T+{A_{\eta }}^{†}{A}_{\eta }{B_{\eta }}^T{\left({A_{\eta }}^{†}\right)}^T+L_{A_{\eta }}U{L_{A_{\eta }}}^T,$$

(5)

where $U=-U^T\in {\mathbb{R}}^{4n\times 4n}$ is an arbitrary matrix.

Proof. 

We only prove the case of $\eta =j$ and the other cases can be conducted in similar ways.

First, we show that any skew symmetric solution to real matrix equation

$$A_j{Y}_j=B_j$$

(6)

can generate a j-anti-Hermitian solution to Equation (1) over ${\mathbb{S}}_g$.

Let us suppose that Equation (6) has a skew symmetric solution Y. Applying (4) of Proposition 1 to Equation (6), we obtain the following three equations:

$$A^{{\sigma }_j}{R}_n{V}_n{G_n}^{-1}Y{R_n}^{-1}=B^{{\sigma }_j},A^{{\sigma }_j}{S}_n{V}_n{G_n}^{-1}Y{S_n}^{-1}=B^{{\sigma }_j},A^{{\sigma }_j}{T}_n{V}_n{G_n}^{-1}Y{T_n}^{-1}=B^{{\sigma }_j}.$$

By direct computation, we have

$$R_n{V}_n=-V_n{R}_n,R_n{G_n}^{-1}=-{G_n}^{-1}{R_n}^T,$$

(7)

$$S_n{V}_n=-V_n{S}_n,S_n{G_n}^{-1}={G_n}^{-1}{S_n}^T,$$

(8)

$$T_n{V}_n=V_n{T}_n,T_n{G_n}^{-1}=-{G_n}^{-1}{T_n}^T,$$

(9)

$${R_n}^{-1}=\frac{1}{\alpha }{R}_n,{S_n}^{-1}=S_n,{T_n}^{-1}=\frac{1}{\alpha }{T}_n.$$

It is easy to show that ${R_n}^{T}Y{R_n}^{-1}$, $-{S_n}^{T}Y{S_n}^{-1}$ and $-{T_n}^{T}Y{T_n}^{-1}$ are skew symmetric solutions to Equation (6). Let

$$\mathcal{Y}=\frac{1}{4}(Y+{R_n}^{T}Y{R_n}^{-1}-{S_n}^{T}Y{S_n}^{-1}-{T_n}^{T}Y{T_n}^{-1}),$$

then, $\mathcal{Y}$ is also a skew symmetric solution to Equation (6).

Assume that Y has the form:

$$Y=\left(\begin{array}{cccc}{Y}_{11}& Y_{12}& Y_{13}& Y_{14}\\ Y_{21}& Y_{22}& Y_{23}& Y_{24}\\ Y_{31}& Y_{32}& Y_{33}& Y_{34}\\ Y_{41}& Y_{42}& Y_{43}& Y_{44}\end{array}\right),Y_{ij}\in {\mathbb{R}}^{n\times n},i,j=1,2,3,4,$$

then, we have

$${G_n}^{-1}\mathcal{Y}=\left(\begin{array}{cccc}-Z_0& -\alpha Z_1& -Z_2& -\alpha Z_3\\ Z_1& Z_0& Z_3& Z_2\\ Z_2& \alpha Z_3& Z_0& \alpha Z_1\\ -Z_3& -Z_2& -Z_1& -Z_0\end{array}\right),$$

where

$$\begin{array}{c}\hfill Z_0=\frac{1}{4}(-Y_{11}-\frac{1}{\alpha }{Y}_{22}+Y_{33}+\frac{1}{\alpha }{Y}_{44}),Z_1=\frac{1}{4\alpha }(-Y_{12}-Y_{21}+Y_{34}+Y_{43}),\\ \hfill Z_2=\frac{1}{4}(-\frac{1}{\alpha }{Y}_{24}+\frac{1}{\alpha }{Y}_{42}+Y_{31}-Y_{13}),Z_3=\frac{1}{4\alpha }(-Y_{14}+Y_{41}-Y_{23}+Y_{32}).\end{array}$$

It is easy to find that ${G_n}^{-1}\mathcal{Y}$ is the j-real representation of the matrix over ${\mathbb{S}}_g$, so by (6) of Proposition 1, we can construct a new matrix X over ${\mathbb{S}}_g$:

$$\begin{array}{cc}\hfill X& =\frac{1}{4}\left(\begin{array}{cccc}-I_n& i{I}_n& j{I}_n& -k{I}_n\end{array}\right){G_n}^{-1}\mathcal{Y}\left(\begin{array}{c}{I}_n\\ \frac{1}{\alpha }i{I}_n\\ j{I}_n\\ \frac{1}{\alpha }k{I}_n\end{array}\right)\hfill \\ \hfill & =\frac{1}{16}\left(\begin{array}{cccc}-I_n& -\frac{1}{\alpha }i{I}_n& j{I}_n& \frac{1}{\alpha }k{I}_n\end{array}\right)(Y+{R_n}^{T}Y{R_n}^{-1}-{S_n}^{T}Y{S_n}^{-1}-{T_n}^{T}Y{T_n}^{-1})\left(\begin{array}{c}{I}_n\\ \frac{1}{\alpha }i{I}_n\\ j{I}_n\\ \frac{1}{\alpha }k{I}_n\end{array}\right).\hfill \end{array}$$

Evidently, ${G_n}^{-1}\mathcal{Y}$ is the j-real representation of X. Note that $\mathcal{Y}$ is a skew symmetric solution to Equation (6). By Proposition 1, we obtain

$$\begin{array}{c}\hfill {\left(AX\right)}^{{\sigma }_j}=A_j\mathcal{Y}=B_j=B^{{\sigma }_j},{\left(X^{j*}\right)}^{{\sigma }_j}={G_n}^{-1}{\mathcal{Y}}^T=-{G_n}^{-1}\mathcal{Y}=-X^{{\sigma }_j},\end{array}$$

clearly, X is a j-anti-Hermitian solution to Equation (1) over ${\mathbb{S}}_g$. Therefore, any skew symmetric solution to Equation (6) can generate a j-anti-Hermitian solution to Equation (1).

Conversely, let us suppose that Equation (1) has a j-anti-Hermitian solution X over ${\mathbb{S}}_g$, by Proposition 1, we obtain

$$A_j\left(G_n{X}^{{\sigma }_j}\right)={\left(AX\right)}^{{\sigma }_j}=B^{{\sigma }_j}=B_j,G_n{X}^{{\sigma }_j}=G_n{(-X^{j*})}^{{\sigma }_j}=-{\left(G_n{X}^{{\sigma }_j}\right)}^T.$$

Thus, $G_n{X}^{{\sigma }_j}$ is a skew symmetric solution to Equation (6). Hence, any j-anti Hermitian solution to Equation (1) can generate a skew symmetric solution to Equation (6).

It is easy to see that Equation (1) has a j-anti-Hermitian solution if and only if the Equation (6) has a skew symmetric solution. By Lemma 1, Equation (6) has a skew symmetric solution $Y_j$ as shown in Formula (5) if and only if condition (3) holds. In this case, by substituting $Y_j$ into Formula (4), we can obtain a j-anti-Hermitian solution to Equation (1). □

3.2. $\eta$-Anti-Hermitian Solution to the System (2)

Lemma 2. 

[27] Let $A\in {\mathbb{R}}^{m\times n}$, $b\in {\mathbb{R}}^m$. Then, the equation $Ax=b$ is consistent if and only if

$$A{A}^{†}b=b.$$

In which case, the general solution is $x=A^{†}b+L_{A}u$, where $u\in {\mathbb{R}}^n$ is an arbitrary vector.

Lemma 3. 

Let us suppose that $A\in {\mathbb{R}}^{k\times n},B\in {\mathbb{R}}^{n\times l},C\in {\mathbb{R}}^{k\times s},D\in {\mathbb{R}}^{s\times l},E\in {\mathbb{R}}^{k\times l}$,

$$\begin{array}{c}\hfill Z_{ij}=\frac{\sqrt{2}}{2}({e_i}^{\left(n\right)}{\left({e_j}^{\left(n\right)}\right)}^T-{e_j}^{\left(n\right)}{\left({e_i}^{\left(n\right)}\right)}^T),i=1,2,\cdots ,n-1;j=i+1,\cdots ,n,\end{array}$$

(10)

$$\begin{array}{c}\hfill W_{pq}=\frac{\sqrt{2}}{2}({e_p}^{\left(s\right)}{\left({e_q}^{\left(s\right)}\right)}^T-{e_q}^{\left(s\right)}{\left({e_p}^{\left(s\right)}\right)}^T),p=1,2,\cdots ,s-1;q=p+1,\cdots ,s,\end{array}$$

(11)

where ${e_i}^{\left(n\right)}$ is the i-th column vector of $I_n$.

Set

$$\begin{array}{c}\hfill S_1=\left(\begin{array}{cccccccc}{Z}_{12}& \cdots & Z_{1n}& Z_{23}& \cdots & Z_{2n}& \cdots & Z_{n-1,n}\end{array}\right)\in {\mathbb{R}}^{n\times \frac{1}{2}{n}^2(n-1)},\\ \hfill S_2=\left(\begin{array}{cccccccc}{W}_{12}& \cdots & W_{1s}& W_{23}& \cdots & W_{2s}& \cdots & W_{s-1,s}\end{array}\right)\in {\mathbb{R}}^{s\times \frac{1}{2}{s}^2(s-1)},\end{array}$$

$$\begin{array}{c}\hfill G=\left(\begin{array}{cccccccc}Vec\left(Z_{12}\right)& \cdots & Vec\left(Z_{1n}\right)& Vec\left(Z_{23}\right)& \cdots & Vec\left(Z_{2n}\right)& \cdots & Vec\left(Z_{n-1,n}\right)\end{array}\right)\in {\mathbb{R}}^{n^2\times \frac{1}{2}n(n-1)}\end{array},$$

(12)

$$\begin{array}{c}\hfill H=\left(\begin{array}{cccccccc}Vec\left(W_{12}\right)& \cdots & Vec\left(W_{1s}\right)& Vec\left(W_{23}\right)& \cdots & Vec\left(W_{2s}\right)& \cdots & Vec\left(W_{s-1,s}\right)\end{array}\right)\in {\mathbb{R}}^{s^2\times \frac{1}{2}s(s-1)},\end{array}$$

(13)

$$M=(B^T\otimes A)G,N=(D^T\otimes C)H,d=Vec\left(E\right),Q=\left(\begin{array}{cc}M& N\end{array}\right).$$

(14)

Then, the real matrix equation

$$AXB+CYD=E$$

(15)

has a skew symmetric solution $(X,Y)$ if and only if

$$Q{Q}^{†}d=d.$$

(16)

In this case, the general skew symmetric solution to Equation(15)can be expressed as

$$X=S_1(a\otimes I_n),Y=S_2(b\otimes I_s),$$

(17)

where

$$a=\left(\begin{array}{cc}{I}_{\frac{1}{2}n(n-1)}& 0\end{array}\right)\sigma ,b=\left(\begin{array}{cc}0& I_{\frac{1}{2}s(s-1)}\end{array}\right)\sigma ,\sigma =Q^{†}d+L_{Q}u,$$

(18)

and $u\in {\mathbb{R}}^{\frac{1}{2}n(n-1)+\frac{1}{2}s(s-1)}$ is an arbitrary vector.

Proof. 

From the definition of $\left\{Z_{ij}\right\}$ and $\left\{W_{pq}\right\}$ in (10) and (11), it is easy to verify that $\left\{Z_{ij}\right\}$ and $\left\{W_{pq}\right\}$ form the orthonormal bases of the set of all skew symmetric matrices in ${\mathbb{R}}^{n\times n}$ and ${\mathbb{R}}^{s\times s}$, respectively. That is,

$$(Z_{ij},Z_{pq})=\left\{\begin{array}{c}0,i\ne porj\ne q\hfill \\ 1,i=pandj=q\hfill \end{array}\right.,(W_{ij},W_{pq})=\left\{\begin{array}{c}0,i\ne porj\ne q\hfill \\ 1,i=pandj=q\hfill \end{array}\right..$$

Now, if X and Y are skew symmetric matrices in ${\mathbb{R}}^{n\times n}$ and ${\mathbb{R}}^{s\times s}$, respectively, they can be expressed as

$$X=\sum _{i,j}^{}{a}_{ij}{Z}_{ij},Y=\sum _{p,q}^{}{b}_{pq}{W}_{pq},$$

(19)

where the real numbers $a_{ij}$ ($i=1,\cdots ,n-1$, $j=i+1,\cdots ,n$) and $b_{pq}$ ($p=1,\cdots ,s-1$, $q=p+1,\cdots ,s$) are yet to be determined. Substitute Formula (19) into Equation (15), we obtain

$$\sum _{i,j}^{}{a}_{ij}A{Z}_{ij}B+\sum _{p,q}^{}{b}_{pq}C{W}_{pq}D=E.$$

(20)

Set

$$a={\left(\begin{array}{cccccccc}{a}_{12}& \cdots & a_{1n}& a_{23}& \cdots & a_{2n}& \cdots & a_{n-1,n}\end{array}\right)}^T,$$

$$b={\left(\begin{array}{cccccccc}{b}_{12}& \cdots & b_{1s}& b_{23}& \cdots & b_{2s}& \cdots & b_{s-1,s}\end{array}\right)}^T,\sigma ={\left(\begin{array}{cc}{a}^T& b^T\end{array}\right)}^T,$$

and let G, H, M, N, d and Q be defined in (12)–(14). Applying the stretching function to both sides of Equation (20), by direct computation, we have

$$Ma+Nb=d.$$

(21)

From the definition of the stretching function, it is evident that Equation (15) is equivalent to Equation (21), which is then equivalent to the following equation:

$$Q\sigma =d.$$

(22)

It follows from Lemma 2 that Equation (22) is consistent if and only if condition (16) holds, in which case, the general solution $\sigma$ to Equation (22) can be shown in Formula (18). Therefore, the general skew symmetric solution to Equation (15) can be shown in Formulas (17) and (18). □

Theorem 3. 

Suppose $A_1\in {\mathbb{S}}_g^{m_1\times n}$, $B_1\in {\mathbb{S}}_g^{m_1\times n}$, $A_2\in {\mathbb{S}}_g^{n\times m_2}$, $B_2\in {\mathbb{S}}_g^{n\times m_2}$, $C_1\in {\mathbb{S}}_g^{t_1\times s}$, $D_1\in {\mathbb{S}}_g^{t_1\times s}$, $C_2\in {\mathbb{S}}_g^{s\times t_2}$, $D_2\in {\mathbb{S}}_g^{s\times t_2}$, $E_1\in {\mathbb{S}}_g^{k\times n}$, $E_2\in {\mathbb{S}}_g^{n\times l}$, $F_1\in {\mathbb{S}}_g^{k\times s}$, $F_2\in {\mathbb{S}}_g^{s\times l}$, $H\in {\mathbb{S}}_g^{k\times l}$, $\eta \in \{i,j,k\}$. Let $Z_{ij}$, $W_{pq}$, $S_1$, $S_2$, G, H be given as in Lemma 3 where n and s change to $4n$ and $4s$, respectively.

Set

$${\mathcal{A}}_{\eta }=\left(\begin{array}{c}{A}_1\\ -A_2^{\eta *}\end{array}\right),{\mathcal{B}}_{\eta }=\left(\begin{array}{c}{B}_1\\ B_2^{\eta *}\end{array}\right),{\mathcal{C}}_{\eta }=\left(\begin{array}{c}{C}_1\\ -C_2^{\eta *}\end{array}\right),{\mathcal{D}}_{\eta }=\left(\begin{array}{c}{D}_1\\ D_2^{\eta *}\end{array}\right),$$

$$A_{\eta }=\left\{\begin{array}{cc}{\mathcal{A}}_i^{{\sigma }_i}{G}_n^{-1},& \eta =i\hfill \\ {\mathcal{A}}_j^{{\sigma }_j}{V}_n{G}_n^{-1},& \eta =j\hfill \\ {\mathcal{A}}_k^{{\sigma }_k}{W}_n{G}_n^{-1},& \eta =k\hfill \end{array}\right.,B_{\eta }={{\mathcal{B}}_{\eta }}^{{\sigma }_{\eta }},C_{\eta }=\left\{\begin{array}{cc}{\mathcal{C}}_i^{{\sigma }_i}{G}_s^{-1},& \eta =i\hfill \\ {\mathcal{C}}_j^{{\sigma }_j}{V}_s{G}_s^{-1},& \eta =j\hfill \\ {\mathcal{C}}_k^{{\sigma }_k}{W}_s{G}_s^{-1},& \eta =k\hfill \end{array}\right.,D_{\eta }={{\mathcal{D}}_{\eta }}^{{\sigma }_{\eta }},$$

$${E_1}_{\eta }=\left\{\begin{array}{cc}{E_1}^{{\sigma }_i}{G}_n^{-1},& \eta =i\hfill \\ {E_1}^{{\sigma }_j}{V}_n{G}_n^{-1},& \eta =j\hfill \\ {E_1}^{{\sigma }_k}{W}_n{G}_n^{-1},& \eta =k\hfill \end{array}\right.,{E_2}_{\eta }=\left\{\begin{array}{cc}{E_2}^{{\sigma }_i},& \eta =i\hfill \\ V_n{E_2}^{{\sigma }_j},& \eta =j\hfill \\ W_n{E_2}^{{\sigma }_k},& \eta =k\hfill \end{array}\right.,$$

$${F_1}_{\eta }=\left\{\begin{array}{cc}{F_1}^{{\sigma }_i}{G}_s^{-1},& \eta =i\hfill \\ {F_1}^{{\sigma }_j}{V}_s{G}_s^{-1},& \eta =j\hfill \\ {F_1}^{{\sigma }_k}{W}_s{G}_s^{-1},& \eta =k\hfill \end{array}\right.,{F_2}_{\eta }=\left\{\begin{array}{cc}{F_2}^{{\sigma }_i},& \eta =i\hfill \\ V_s{F_2}^{{\sigma }_j},& \eta =j\hfill \\ W_s{F_2}^{{\sigma }_k},& \eta =k\hfill \end{array}\right.,H_{\eta }=H^{{\sigma }_{\eta }},$$

$${\widehat{E}}_{1_{\eta }}={E_1}_{\eta }{L}_{A_{\eta }},{\widehat{E}}_{2_{\eta }}=L_{A_{\eta }}^T{E_2}_{\eta },{\widehat{F}}_{1_{\eta }}={F_1}_{\eta }{L}_{C_{\eta }},{\widehat{F}}_{2_{\eta }}=L_{C_{\eta }}^T{F_2}_{\eta },$$

(23)

$$\begin{array}{cc}\hfill {\widehat{H}}_{\eta }=H_{\eta }& -{E_1}_{\eta }[{A_{\eta }}^{†}{B}_{\eta }-{\left({A_{\eta }}^{†}{B}_{\eta }\right)}^T+{A_{\eta }}^{†}{A}_{\eta }{B_{\eta }}^T{\left({A_{\eta }}^{†}\right)}^T]{E_2}_{\eta }\hfill \\ \hfill & -{F_1}_{\eta }[{C_{\eta }}^{†}{D}_{\eta }-{\left({C_{\eta }}^{†}{D}_{\eta }\right)}^T+{C_{\eta }}^{†}{C}_{\eta }{D_{\eta }}^T{\left({C_{\eta }}^{†}\right)}^T]{F_2}_{\eta }\hfill \end{array},$$

(24)

$${\widehat{M}}_{\eta }=({\left({\widehat{E}}_{2_{\eta }}\right)}^T\otimes {\widehat{E}}_{1_{\eta }})G,{\widehat{N}}_{\eta }=({\left({\widehat{F}}_{2_{\eta }}\right)}^T\otimes {\widehat{F}}_{1_{\eta }})H,d_{\eta }=Vec\left({\widehat{H}}_{\eta }\right),Q_{\eta }=\left(\begin{array}{cc}{\widehat{M}}_{\eta }& {\widehat{N}}_{\eta }\end{array}\right).$$

(25)

Then, the system (2) has an η-anti-Hermitian solution $(X,Y)$ over ${\mathbb{S}}_g$ if and only if

(1) 

The corresponding system of real matrix equations

$$\begin{array}{c}\hfill \left\{\begin{array}{ccc}\hfill A_{\eta }{M}_{\eta }& =\hfill & B_{\eta }\hfill \\ \hfill C_{\eta }{N}_{\eta }& =\hfill & D_{\eta }\hfill \\ \hfill {E_1}_{\eta }{M}_{\eta }{E_2}_{\eta }+{F_1}_{\eta }{N}_{\eta }{F_2}_{\eta }& =\hfill & H_{\eta }\hfill \end{array}\right.\end{array}$$

(26)

has a skew symmetric solution $(M_{\eta },N_{\eta })$.

(2) 

The following conditions hold:

$$R_{A_{\eta }}{B}_{\eta }=0,B_{\eta }{A_{\eta }}^T=-A_{\eta }{B_{\eta }}^T,R_{C_{\eta }}{D}_{\eta }=0,D_{\eta }{C_{\eta }}^T=-C_{\eta }{D_{\eta }}^T,$$

(27)

$$Q_{\eta }{Q}_{\eta }^{†}{d}_{\eta }=d_{\eta }.$$

(28)

The above two statements are equivalent to each other. In which case, the general η-anti-Hermitian solution to the system(2)can be expressed as follows:

(a) 

in the case of $\eta =i$,

$$\begin{array}{cc}\hfill X& =\frac{1}{16}\left(\begin{array}{cccc}{I}_n& -\frac{1}{\alpha }i{I}_n& j{I}_n& -\frac{1}{\alpha }k{I}_n\end{array}\right)(M_i-{R_n}^T{M}_i{R_n}^{-1}+{S_n}^T{M}_i{S_n}^{-1}-{T_n}^T{M}_i{T_n}^{-1})\left(\begin{array}{c}{I}_n\\ \frac{1}{\alpha }i{I}_n\\ j{I}_n\\ \frac{1}{\alpha }k{I}_n\end{array}\right),\hfill \\ \hfill Y& =\frac{1}{16}\left(\begin{array}{cccc}{I}_s& -\frac{1}{\alpha }i{I}_s& j{I}_s& -\frac{1}{\alpha }k{I}_s\end{array}\right)(N_i-{R_s}^T{N}_i{R_s}^{-1}+{S_s}^T{N}_i{S_s}^{-1}-{T_s}^T{N}_i{T_s}^{-1})\left(\begin{array}{c}{I}_s\\ \frac{1}{\alpha }i{I}_s\\ j{I}_s\\ \frac{1}{\alpha }k{I}_s\end{array}\right),\hfill \end{array}$$

(b) 

in the case of $\eta =j$,

$$\begin{array}{c}\hfill X=\frac{1}{16}\left(\begin{array}{cccc}-I_n& -\frac{1}{\alpha }i{I}_n& j{I}_n& \frac{1}{\alpha }k{I}_n\end{array}\right)(M_j+{R_n}^T{M}_j{R_n}^{-1}-{S_n}^T{M}_j{S_n}^{-1}-{T_n}^T{M}_j{T_n}^{-1})\left(\begin{array}{c}{I}_n\\ \frac{1}{\alpha }i{I}_n\\ j{I}_n\\ \frac{1}{\alpha }k{I}_n\end{array}\right),\end{array}$$

(29)

$$Y=\frac{1}{16}\left(\begin{array}{cccc}-I_s& -\frac{1}{\alpha }i{I}_s& j{I}_s& \frac{1}{\alpha }k{I}_s\end{array}\right)(N_j+{R_s}^T{N}_j{R_s}^{-1}-{S_s}^T{N}_j{S_s}^{-1}-{T_s}^T{N}_j{T_s}^{-1})\left(\begin{array}{c}{I}_s\\ \frac{1}{\alpha }i{I}_s\\ j{I}_s\\ \frac{1}{\alpha }k{I}_s\end{array}\right),$$

(30)

(c) 

in the case of $\eta =k$,

$$\begin{array}{cc}\hfill X& =\frac{1}{16}\left(\begin{array}{cccc}{I}_n& -\frac{1}{\alpha }i{I}_n& -j{I}_n& \frac{1}{\alpha }k{I}_n\end{array}\right)(M_k-{R_n}^T{M}_k{R_n}^{-1}-{S_n}^T{M}_k{S_n}^{-1}+{T_n}^T{M}_k{T_n}^{-1})\left(\begin{array}{c}{I}_n\\ \frac{1}{\alpha }i{I}_n\\ j{I}_n\\ \frac{1}{\alpha }k{I}_n\end{array}\right),\hfill \\ \hfill Y& =\frac{1}{16}\left(\begin{array}{cccc}{I}_s& -\frac{1}{\alpha }i{I}_s& -j{I}_s& \frac{1}{\alpha }k{I}_s\end{array}\right)(N_k-{R_s}^T{N}_k{R_s}^{-1}-{S_s}^T{N}_k{S_s}^{-1}+{T_s}^T{N}_k{T_s}^{-1})\left(\begin{array}{c}{I}_s\\ \frac{1}{\alpha }i{I}_s\\ j{I}_s\\ \frac{1}{\alpha }k{I}_s\end{array}\right).\hfill \end{array}$$

In above (a)–(c),

$$M_{\eta }={A_{\eta }}^{†}{B}_{\eta }-{\left({A_{\eta }}^{†}{B}_{\eta }\right)}^T+{A_{\eta }}^{†}{A}_{\eta }{B_{\eta }}^T{\left({A_{\eta }}^{†}\right)}^T+L_{A_{\eta }}{U}_{\eta }{L_{A_{\eta }}}^T,$$

(31)

$$N_{\eta }={C_{\eta }}^{†}{D}_{\eta }-{\left({C_{\eta }}^{†}{D}_{\eta }\right)}^T+{C_{\eta }}^{†}{C}_{\eta }{D_{\eta }}^T{\left({C_{\eta }}^{†}\right)}^T+L_{C_{\eta }}{V}_{\eta }{L_{C_{\eta }}}^T,$$

(32)

where

$$U_{\eta }=S_1(a_{\eta }\otimes I_{4n}),V_{\eta }=S_2(b_{\eta }\otimes I_{4s}),$$

(33)

$$a_{\eta }=\left(\begin{array}{cc}{I}_{2n(4n-1)}& 0\end{array}\right){\sigma }_{\eta },b_{\eta }=\left(\begin{array}{cc}0& I_{2s(4s-1)}\end{array}\right){\sigma }_{\eta },{\sigma }_{\eta }=Q_{\eta }^{†}{d}_{\eta }+L_{Q_{\eta }}u,$$

(34)

and $u\in {\mathbb{R}}^{2n(4n-1)+2s(4s-1)}$ is an arbitrary vector.

Proof. 

We only prove the case of $\eta =j$ and the other cases can be conducted in similar ways.

At first, it is clear that the following system of matrix equations

$$\left\{\begin{array}{ccc}{A}_{1}X\hfill & =\hfill & B_1\hfill \\ X{A}_2\hfill & =\hfill & B_2\hfill \end{array}\right.$$

has a j-anti-Hermitian solution X over ${\mathbb{S}}_g$ if and only if

$$\left\{\begin{array}{ccc}\hfill A_{1}X& =\hfill & B_1\hfill \\ \hfill -A_2^{j*}X& =\hfill & B_2^{j*}\hfill \end{array}\right.i.e.,{\mathcal{A}}_{j}X={\mathcal{B}}_j$$

has a j-anti-Hermitian solution X over ${\mathbb{S}}_g$, by Theorem 2, if and only if

$$A_j{M}_j=B_j$$

has a skew symmetric solution M over $\mathbb{R}$. Similarly,

$$\left\{\begin{array}{ccc}\hfill A_{1}X& =\hfill & B_1\hfill \\ \hfill X{A}_2& =\hfill & B_2\hfill \\ \hfill C_{1}Y& =\hfill & D_1\hfill \\ \hfill Y{C}_2& =\hfill & D_2\hfill \end{array}\right.$$

has a j-anti-Hermitian solution $(X,Y)$ over ${\mathbb{S}}_g$ if and only if

$$\left\{\begin{array}{ccc}\hfill A_j{M}_j& =\hfill & B_j\hfill \\ \hfill C_j{N}_j& =\hfill & D_j\hfill \end{array}\right.$$

(35)

has a skew symmetric solution $(M,N)$ over $\mathbb{R}$.

Then, we show that any skew symmetric solution $(M,N)$ to the system

$$\begin{array}{c}\hfill \left\{\begin{array}{ccc}\hfill A_j{M}_j& =\hfill & B_j\hfill \\ \hfill C_j{N}_j& =\hfill & D_j\hfill \\ \hfill {E_1}_j{M}_j{E_2}_j+{F_1}_j{N}_j{F_2}_j& =\hfill & H_j\hfill \end{array}\right.\end{array}$$

(36)

can generate a j-anti-Hermitian solution $(X,Y)$ to the system (2).

Let us suppose that the system (36) has a skew symmetric solution $(M,N)$. By the proof of Theorem 2,

$$\mathcal{M}=\frac{1}{4}(M+{R_n}^{T}M{R_n}^{-1}-{S_n}^{T}M{S_n}^{-1}-{T_n}^{T}M{T_n}^{-1})$$

and

$$\mathcal{N}=\frac{1}{4}(N+{R_s}^{T}N{R_s}^{-1}-{S_s}^{T}N{S_s}^{-1}-{T_s}^{T}N{T_s}^{-1})$$

are also the skew symmetric solutions to the system (35). Apply (4) of Proposition 1 to

$${E_1}_j{M}_j{E_2}_j+{F_1}_j{N}_j{F_2}_j=H_j.$$

(37)

By (7)–(9), we obtain the following three equations

$$\begin{array}{cc}\hfill {E_1}_j\left({R_n}^{T}M{R_n}^{-1}\right){E_2}_j+{F_1}_j\left({R_s}^{T}N{R_s}^{-1}\right){F_2}_j& =H_j,\hfill \\ \hfill {E_1}_j(-{S_n}^{T}M{S_n}^{-1}){E_2}_j+{F_1}_j(-{S_s}^{T}N{S_s}^{-1}){F_2}_j& =H_j,\hfill \\ \hfill {E_1}_j(-{T_n}^{T}M{T_n}^{-1}){E_2}_j+{F_1}_j(-{T_s}^{T}N{T_s}^{-1}){F_2}_j& =H_j.\hfill \end{array}$$

Hence, $\mathcal{M}$ and $\mathcal{N}$ are also the skew symmetric solutions to Equation (37) and therefore to the system (36).

From the proof of Theorem 2, we know that ${G_n}^{-1}\mathcal{M}$ and ${G_s}^{-1}\mathcal{N}$ are the j-real representations of matrices over ${\mathbb{S}}_g$; moreover, we can construct new matrices X and Y over ${\mathbb{S}}_g$:

$$\begin{array}{cc}\hfill X& =\frac{1}{4}\left(\begin{array}{cccc}-I_n& i{I}_n& j{I}_n& -k{I}_n\end{array}\right){G_n}^{-1}\mathcal{M}\left(\begin{array}{c}{I}_n\\ \frac{1}{\alpha }i{I}_n\\ j{I}_n\\ \frac{1}{\alpha }k{I}_n\end{array}\right)\hfill \\ \hfill & =\frac{1}{16}\left(\begin{array}{cccc}-I_n& -\frac{1}{\alpha }i{I}_n& j{I}_n& \frac{1}{\alpha }k{I}_n\end{array}\right)(M+{R_n}^{T}M{R_n}^{-1}-{S_n}^{T}M{S_n}^{-1}-{T_n}^{T}M{T_n}^{-1})\left(\begin{array}{c}{I}_n\\ \frac{1}{\alpha }i{I}_n\\ j{I}_n\\ \frac{1}{\alpha }k{I}_n\end{array}\right),\hfill \end{array}$$

$$\begin{array}{cc}\hfill Y& =\frac{1}{4}\left(\begin{array}{cccc}-I_s& i{I}_s& j{I}_s& -k{I}_s\end{array}\right){G_s}^{-1}\mathcal{N}\left(\begin{array}{c}{I}_s\\ \frac{1}{\alpha }i{I}_s\\ j{I}_s\\ \frac{1}{\alpha }k{I}_s\end{array}\right)\hfill \\ \hfill & =\frac{1}{16}\left(\begin{array}{cccc}-I_s& -\frac{1}{\alpha }i{I}_s& j{I}_s& \frac{1}{\alpha }k{I}_s\end{array}\right)(N+{R_s}^{T}N{R_s}^{-1}-{S_s}^{T}N{S_s}^{-1}-{T_s}^{T}N{T_s}^{-1})\left(\begin{array}{c}{I}_s\\ \frac{1}{\alpha }i{I}_s\\ j{I}_s\\ \frac{1}{\alpha }k{I}_s\end{array}\right).\hfill \end{array}$$

Clearly, ${G_n}^{-1}\mathcal{M}$ and ${G_s}^{-1}\mathcal{N}$ are the j-real representations of X and Y, respectively. According to Theorem 2, $(X,Y)$ is a j-anti-Hermitian solution to the system (35) over ${\mathbb{S}}_g$.

Note that $(\mathcal{M},\mathcal{N})$ is a skew symmetric solution to Equation (37), we obtain

$${\left(E_{1}X{E}_2\right)}^{{\sigma }_j}+{\left(F_{1}Y{F}_2\right)}^{{\sigma }_j}={E_1}_j\mathcal{M}{E_2}_j+{F_1}_j\mathcal{N}{F_2}_j=H_j=H^{{\sigma }_j},$$

i.e.,

$$E_{1}X{E}_2+F_{1}Y{F}_2=H.$$

(38)

Hence, $(X,Y)$ is a j-anti-Hermitian solution to Equation (38) and therefore to the system (2). Consequently, any skew symmetric solution $(M,N)$ to the system (36) can generate a j-anti-Hermitian solution to the system (2).

Conversely, let us suppose that the system (2) has a j-anti-Hermitian solution $(X,Y)$ over ${\mathbb{S}}_g$, then we obtain

$$A_j\left(G_n{X}^{{\sigma }_j}\right)={\left({\mathcal{A}}_{j}X\right)}^{{\sigma }_j}={{\mathcal{B}}_j}^{{\sigma }_j}=B_j,C_j\left(G_s{Y}^{{\sigma }_j}\right)={\left({\mathcal{C}}_{j}Y\right)}^{{\sigma }_j}={{\mathcal{D}}_j}^{{\sigma }_j}=D_j,$$

$${E_1}_j\left(G_n{X}^{{\sigma }_j}\right){E_2}_j+{F_1}_j\left(G_s{Y}^{{\sigma }_j}\right){F_2}_j={\left(E_{1}X{E}_2\right)}^{{\sigma }_j}+{\left(F_{1}Y{F}_2\right)}^{{\sigma }_j}=H^{{\sigma }_j}=H_j.$$

Thus, ($G_n{X}^{{\sigma }_j}$,$G_s{Y}^{{\sigma }_j}$) is a skew symmetric solution to the system (36) over $\mathbb{R}$, then any j-anti-Hermitian solution to the system (2) can generate a skew symmetric solution to the system (36).

It is clear that the system (2) has a j-anti-Hermitian solution if and only if the system (36) has a skew symmetric solution. By Lemma 1, the system (35) has a skew symmetric solution if and only if condition (27) holds, the general skew symmetric solutions ($M_j$,$N_j$) to the system (35) are as shown in Formulas (31) and (32), where $U_j=-{U_j}^T\in {\mathbb{R}}^{4n\times 4n}$ and $V_j=-{V_j}^T\in {\mathbb{R}}^{4s\times 4s}$ are arbitrary matrices.

By substituting Formulas (31) and (32) into Equation (37), we have

$${\widehat{E}}_{1_j}{U}_j{\widehat{E}}_{2_j}+{\widehat{F}}_{1_j}{V}_j{\widehat{F}}_{2_j}={\widehat{H}}_j,$$

(39)

where ${\widehat{E}}_{1_j}$, ${\widehat{E}}_{2_j}$, ${\widehat{F}}_{1_j}$, ${\widehat{F}}_{2_j}$ and ${\widehat{H}}_j$ are defined as in (23)–(24).

According to Lemma 3, Equation (39) has a skew symmetric solution $(U_j,V_j)$ if and only if condition (28) holds, where ${\widehat{M}}_j$, ${\widehat{N}}_j$, $d_j$ and $Q_j$ are defined as in Formula (25); then, the general expression of the skew symmetric solution to the Equation (39) as shown in Formulas (33) and (34).

In conclusion, the system (2) has a j-anti-Hermitian solution over ${\mathbb{S}}_g$ if and only if conditions (27) and (28) hold. In this case, the expression of the general j-anti-Hermitian solution to the system (2) can be given by (29)–(34). □

Corollary 1. 

Under the same definitions in Theorem 3, if conditions(27)and(28)hold, and

$$rank\left(Q_{\eta }\right)=2n(4n-1)+2s(4s-1),$$

(40)

then the system of real matrix Equations (26) has a unique skew symmetric solution

$$M_{\eta }=\left(\begin{array}{cccc}{M}_{{\eta }_{11}}& M_{{\eta }_{12}}& M_{{\eta }_{13}}& M_{{\eta }_{14}}\\ M_{{\eta }_{21}}& M_{{\eta }_{22}}& M_{{\eta }_{23}}& M_{{\eta }_{24}}\\ M_{{\eta }_{31}}& M_{{\eta }_{32}}& M_{{\eta }_{33}}& M_{{\eta }_{34}}\\ M_{{\eta }_{41}}& M_{{\eta }_{42}}& M_{{\eta }_{43}}& M_{{\eta }_{44}}\end{array}\right),N_{\eta }=\left(\begin{array}{cccc}{N}_{{\eta }_{11}}& N_{{\eta }_{12}}& N_{{\eta }_{13}}& N_{{\eta }_{14}}\\ N_{{\eta }_{21}}& N_{{\eta }_{22}}& N_{{\eta }_{23}}& N_{{\eta }_{24}}\\ N_{{\eta }_{31}}& N_{{\eta }_{32}}& N_{{\eta }_{33}}& N_{{\eta }_{34}}\\ N_{{\eta }_{41}}& N_{{\eta }_{42}}& N_{{\eta }_{43}}& N_{{\eta }_{44}}\end{array}\right),$$

where $M_{{\eta }_{pq}}\in {\mathbb{R}}^{n\times n}$ and $N_{{\eta }_{pq}}\in {\mathbb{R}}^{s\times s}$. In this case, the system (2) also has a unique η-anti-Hermitian solution $X=X_0+X_{1}i+X_{2}j+X_{3}k$, $Y=Y_0+Y_{1}i+Y_{2}j+Y_{3}k$, which can be given as follows:

(1) 

in the case of $\eta =\{i,k\}$,

$$\begin{array}{ccccccccc}\hfill X_0& =M_{{\eta }_{11}},\hfill & \hfill X_1& =\frac{1}{\alpha }{M}_{{\eta }_{12}},\hfill & \hfill X_2& =M_{{\eta }_{13}},\hfill & \hfill X_3& =\frac{1}{\alpha }{M}_{{\eta }_{14}},\hfill & \\ \hfill Y_0& =N_{{\eta }_{11}},\hfill & \hfill Y_1& =\frac{1}{\alpha }{N}_{{\eta }_{12}},\hfill & \hfill Y_2& =N_{{\eta }_{13}},\hfill & \hfill Y_3& =\frac{1}{\alpha }{N}_{{\eta }_{14}},\hfill \end{array}$$

(2) 

in the case of $\eta =j$,

$$\begin{array}{ccccccccc}\hfill X_0& =-M_{j_{11}},\hfill & \hfill X_1& =-\frac{1}{\alpha }{M}_{j_{12}},\hfill & \hfill X_2& =-M_{j_{13}},\hfill & \hfill X_3& =-\frac{1}{\alpha }{M}_{j_{14}},\hfill & \\ \hfill Y_0& =-N_{j_{11}},\hfill & \hfill Y_1& =-\frac{1}{\alpha }{N}_{j_{12}},\hfill & \hfill Y_2& =-N_{j_{13}},\hfill & \hfill Y_3& =-\frac{1}{\alpha }{N}_{j_{14}}.\hfill \end{array}$$

Proof. 

Since rank equality (40) holds, the column vectors of $Q_j$ are linearly independent. By Lemma 2 in Chapter 1 of [27], we obtain

$${Q_j}^{†}{Q}_j=I_{2n(4n-1)+2s(4s-1)}.$$

In this case, $L_{Q_j}=0$. According to Theorem 3, the system (26) has a unique skew symmetric solution $(M,N)$ and the system (2) has at least one $\eta$-anti-Hermitian solution. Let us assume that the system (2) has two different solutions $(X,Y)$ and $(\widehat{X},\widehat{Y})$. By the proof of Theorem 3, $(G_n{X}^{{\sigma }_{\eta }},G_s{Y}^{{\sigma }_{\eta }})$ and $(G_n{\widehat{X}}^{{\sigma }_{\eta }},G_s{\widehat{Y}}^{{\sigma }_{\eta }})$ are two different skew symmetric solutions to the system (26) which conflicts with our assumption. Therefore, the system (2) has a unique $\eta$-anti-Hermitian solution $(X,Y)$. Let us assume $(M_{\eta },N_{\eta })$ is a unique symmetric solution to the system (26); we can obtain $X_t$ and $Y_t$$(t=0,1,2,3)$ from $X^{{\sigma }_{\eta }}$ and $Y^{{\sigma }_{\eta }}$, where $X^{{\sigma }_{\eta }}=G_n^{-1}{M}_{\eta }$ and $Y^{{\sigma }_{\eta }}=G_s^{-1}{N}_{\eta }$. □

3.3. Numerical Examples

Consider the general j-anti-Hermitian solution to the system (2) over $H_g$ when $\alpha =-1$.

Let

$$A_1=\left(\begin{array}{cc}2-i+4j+k& -i+3j+2k\end{array}\right),B_1=\left(\begin{array}{cc}12+6i& 6+6j-6k\end{array}\right),$$

$$A_2=\left(\begin{array}{c}1-2i+j+k\\ 2-i+j+k\end{array}\right),B_2=\left(\begin{array}{c}6+6j-6k\\ 6i\end{array}\right),C_1=\left(\begin{array}{cc}2+i-j& 1+j+k\end{array}\right),$$

$$D_1=\left(\begin{array}{cc}-k& 1+2i+j\end{array}\right),C_2=\left(\begin{array}{c}1+i+j+2k\\ -1-i+j\end{array}\right),D_2=\left(\begin{array}{c}-1-2i-j\\ 2j-k\end{array}\right),$$

$$E_1=\left(\begin{array}{cc}j+k& 1+i\\ 2+2j& 2i+j+k\\ 1+2j+k& 3k\end{array}\right),E_2=\left(\begin{array}{ccc}i-5j& 3+2k& 1+2i+5k\\ -2-k& 6j& 3i+j+3k\end{array}\right),$$

$$F_1=\left(\begin{array}{cc}6-j& 3i-6k\\ i+2k& -5\\ 3+k& -2i+3j\end{array}\right),F_2=\left(\begin{array}{ccc}i-j+2k& 3+k& 2j\\ -5& -2i+3j& 6+3i+j-6k\end{array}\right),$$

$$H=\left(\begin{array}{ccc}-53-69i-48j+3k& 62-19i+15j+45k& -35+80i+152j-37k\\ -41-42i-67j-37k& 41+53i+27j+42k& 21+84i+54j+153k\\ -15+5i-83j-43k& 86+30i-44j+18k& 42+34i-3j+81k\end{array}\right).$$

By direct computation of $A_j$, $B_j$, $C_j$, $D_j$, $Q_j$ and $d_j$, it can be verified that

$$R_{A_j}{B}_j=0,B_j{A_j}^T=-A_j{B_j}^T,R_{C_j}{D}_j=0,D_j{C_j}^T=-C_j{D_j}^T,Q_j{Q}_j^{†}{d}_j=d_j,$$

and

$$rank\left(Q_j\right)=56.$$

According to Corollary 1, the system (26) has a unique skew symmetric solution. By computation, we obtain the unique j-anti-Hermitian solution to the system (2) over $H_g$ is

$$X=\left(\begin{array}{cc}3j+3k& 2-j-3k\\ -2-j-3k& 3j+3k\end{array}\right),Y=\left(\begin{array}{cc}k& 2+i+2j\\ -2-i+2j& -j-k\end{array}\right).$$

4. Conclusions

In this paper, we discuss the existence and general expression of the $\eta$-anti-Hermitian solution to the generalized Segre quaternion matrix Equation (1) and the system (2) by using the real representations of matrices over the generalized Segre quaternion algebra, Moore–Penrose generalized inverse, Kronecker product and the stretching function. In the end, a numerical example is given to verify the main results. The $\eta$-Hermitian solution to the generalized Segre quaternion matrix Equation (1) and the system (2) and other systems may be considered in the future.