A System of Coupled Matrix Equations with an Application over the Commutative Quaternion Ring

Abstract

In this paper, we study the necessary and sufficient conditions for a system of matrix equations to have a solution and a Hermitian solution. As an application, we establish the necessary and sufficient conditions for a classical matrix system to have a reducible solution. Finally, we present an algorithm, along with two concrete examples to validate the main conclusions.

1. Introduction

In this study, we investigated the following coupled matrix equations to determine their solvability over the commutative quaternion ring:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{A}_1{X}_1{B}_1+A_1{X}_2{B}_2+A_2{X}_3{B}_2=C_1,\\ E_1{X}_1{F}_1+E_1{X}_2{F}_2+E_2{X}_3{F}_2=C_2,\\ G_1{X}_1{H}_1+G_1{X}_2{H}_2+G_2{X}_3{H}_2=C_3.\end{array}\right.\end{array}$$

(1)

We also derived the general solution for the system when it is consistent. Additionally, we examined the conditions under which the system allows for a Hermitian solution and provided an explicit formula for such solutions. As an application of System (1), we derived the reducible solution of the following classical matrix system within the commutative quaternion ring:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}AXB=C_1,\\ EXF=C_2,\\ GXH=C_3.\end{array}\right.\end{array}$$

(2)

Commutative quaternions, which fulfill the multiplication commutative rule, form a ring within four-dimensional Clifford algebra. The concept of commutative quaternions was proposed by Segre [1] in 1892. The set of all commutative quaternions is defined as follows:

$${\mathbb{Q}}_c=\{m=m_0+m_1\mathbf{i}+m_2\mathbf{j}+m_3\mathbf{k}|m_0,m_1,m_2,m_3\in \mathbb{R}\},$$

where the imaginary units $\mathbf{i}$, $\mathbf{j}$, and $\mathbf{k}$ satisfy the following conditions:

$${\mathbf{i}}^2={\mathbf{k}}^2=-1,{\mathbf{j}}^2=1,\mathbf{ijk}=-1,\mathbf{ij}=\mathbf{ji}=\mathbf{k},\mathbf{jk}=\mathbf{kj}=\mathbf{i},\mathbf{ki}=\mathbf{ik}=-\mathbf{j}.$$

The utilization of commutative quaternion algebra is extensive, encompassing applications in signal and image processing, as well as Hopfield neural networks (e.g., [2,3,4,5,6,7,8,9]).

Linear matrix equations remain a highly active area of research in mathematics, with wide-ranging applications in domains such as neural networks [10,11] and descriptor systems in control theory [12]. The extensive literature on this topic underscores its importance (see, e.g., [13,14,15,16,17,18,19,20,21,22]). For instance, Xie and Wang [13] considered some equivalent conditions for the following quaternion equation to be solvable:

$$A_1{X}_1{B}_1+A_1{X}_2{B}_2+A_2{X}_3{B}_2=C_1,$$

(3)

Liu and Zhang [14] derived the necessary and sufficient conditions for the consistency of the following coupled quaternion matrix equations:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{E}_1{X}_1{F}_1+E_1{X}_2{F}_2+E_2{X}_3{F}_2=D_1,\\ G_1{Y}_1{H}_1+G_1{Y}_2{H}_2+G_2{X}_3{H}_2=D_2.\end{array}\right.\end{array}$$

(4)

He and Wang [15] investigated some solvability conditions for the following system over quaternion algebra:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}AXB=C_1,\\ EXF=C_2,\\ GXH=C_3.\end{array}\right.\end{array}$$

(5)

However, there is limited information regarding the study of systems, such as System (5) involving reducible solutions using commutative quaternions. In this study, we investigated the equivalent conditions for a system (such as System (5)) to be reducible over commutative quaternions. A square quaternion matrix Z is said to be reducible if there exists a permutation matrix K so that

$$Z=K\left(\begin{array}{cc}{Z}_1& Z_2\\ 0& Z_3\end{array}\right)K^{-1},$$

where $Z_1$ and $Z_2$ represent square matrices with appropriate dimensions. If the order of $Z_3$ is k, we describe Z as k-reducible with respect to permutation matrix K. For any but a fixed permutation matrix K, we substitute

$${\mathbb{Q}}_k^{m\times m}=\left\{\begin{array}{c}Z=K\left(\begin{array}{cc}{Z}_1& Z_2\\ 0& Z_3\end{array}\right)K^{-1}|1\le k<m,Z_1\in {\mathbb{Q}}_c^{(m-k)\times (m-k)},Z_3\in {\mathbb{Q}}_c^{k\times k}\end{array}\right\},$$

where K is any but a fixed permutation matrix. Reducible matrices have well-established applications in numerous fields, including the connection of compartmental analyses, directed graphs, biology, stochastic processes, and continuous-time positive systems (see, e.g., [23,24,25,26,27]).

Hermitian matrices have an extensive array of applications in numerical analysis, information and linear system theory, and engineering problems. Therefore, a significant body of research has focused on Hermitian solutions to matrix and operator equations, such as refs. [28,29,30,31,32]. Ren, Wang, and Chen [33] investigated the $\eta$-anti-Hermitian solutions for a system of constrained matrix equations over the generalized commutative quaternion algebra. Zhang, Wang, and Xie [34] studied the Hermitian solutions of a new system of commutative quaternion matrix equations.

Motivated by the aforementioned work, the widespread utilization of commutative quaternions, the importance of linear matrix equations, and the development of theory, in this study, we considered a solvability condition for the commutative quaternion matrix System (1) and a general expression of System (1) when it has a solution. Accordingly, we derived an equivalent condition for System (1) to obtain a Hermitian solution, along with a formula for the solution. Subsequently, as an application, we explored the reducible solutions of classical matrix equations (2) over the commutative quaternion ring.

The remainder of this article is structured as follows: In Section 2, we introduce the necessary definitions and lemmas. In Section 3, we establish the equivalent condition for System (1) to be solvable over the commutative quaternion ring and obtain a general solution expression for System (1). In Section 4, we study the Hermitian solution of System (1) over the commutative quaternion ring and derive the Hermitian solution expression of System (1). In Section 5, as an application of Equation (1), we study the reducible solutions for classical matrix equations (2) over the commutative quaternion ring. In Section 6, we present an algorithm, along with a corresponding example to demonstrate the principal findings of this study. Finally, we conclude the article with a concise summary in Section 7.

2. Preliminaries

This section presents the key definitions and lemmas underpinning the arguments and proofs throughout the article.

In this manuscript, the symbols $\mathbb{R}$, $\mathbb{C}$, and ${\mathbb{Q}}_c$ are employed to denote the real number field, the complex number field, and the commutative quaternion ring, respectively. Furthermore, ${\mathbb{R}}^{m\times n}$, ${\mathbb{C}}^{m\times n}$, and ${\mathbb{Q}}_c^{m\times n}$ are utilized to represent the sets of all $m\times n$ matrices over $\mathbb{R}$, $\mathbb{C}$, and ${\mathbb{Q}}_c$, respectively. The notations ${\mathbb{ASR}}^{n\times n}$, ${\mathbb{SR}}^{n\times n}$, and ${\mathbb{HQ}}_c^{n\times n}$ are designated to signify the collections of all $n\times n$ real anti-symmetric matrices, $n\times n$ real symmetric matrices, and $n\times n$ Hermitian commutative quaternion matrices, respectively. The detail information of the symbols in the paper can be found in Appendix A.

Let $A\in {\mathbb{Q}}_c^{m\times n}$ be a commutative quaternion matrix. We denote its conjugate transpose and standard transpose by $A^{*}$ and $A^T$, respectively. For a complex matrix $B\in {\mathbb{C}}^{m\times n},$ we define its real and imaginary components through the following decomposition:

$$\begin{array}{c}\hfill B=Re\left(B\right)+\mathbf{i}Im\left(B\right),\end{array}$$

where $Re\left(B\right):={\displaystyle \frac{1}{2}}(B+\overline{B})$ and $Im\left(B\right):={\displaystyle \frac{1}{2\mathbf{i}}}(B-\overline{B})$, with $\overline{B}$ denoting complex conjugation. The Moore–Penrose inverse of a matrix $B\in {\mathbb{Q}}_c^{m\times n}$, denoted as $B^{†}$, is the unique matrix $M\in {\mathbb{Q}}_c^{n\times m}$ satisfying $BMB=B,MBM=M,{\left(BM\right)}^{*}=BM,{\left(MB\right)}^{*}=MB$.

Definition 1 

([35]). Let $D=D_1+D_2\mathbf{j}\in {\mathbb{Q}}_c^{m\times n}$ be given, where $D_1,D_2\in {\mathbb{C}}^{m\times n}$. The complex representation of D is defined as follows:

$$S\left(D\right):=\left(\begin{array}{cc}{D}_1& D_2\\ D_2& D_1\end{array}\right).$$

Definition 2 

([35]). For any $D=D_1+D_2\mathbf{j}\in {\mathbb{Q}}_c^{m\times n},D_1,D_2\in {\mathbb{C}}^{m\times n},$ the following equations hold:

$$\begin{array}{cc}\hfill & D_1+D_2\mathbf{j}=D\cong {\psi }_D=\left(\begin{array}{cc}{D}_1& D_2\end{array}\right),\hfill \\ \hfill & \widehat{D_1}=\left(\begin{array}{c}Re\left(D_1\right)\\ Im\left(D_1\right)\end{array}\right),\widehat{D}=\left(\begin{array}{c}Re\left(D_1\right)\\ Im\left(D_1\right)\\ Re\left(D_2\right)\\ Im\left(D_2\right)\end{array}\right),\hfill \end{array}$$

The vec-operator of $D=\left(d_{ij}\right)\in {\mathbb{Q}}_c^{m\times n}$ is defined as follows:

$$\mathrm{vec}\left(D\right)={\left(d_1,d_2,\dots ,d_n\right)}^T,d_j=\left(d_{1j},d_{2j},\dots ,d_{mj}\right),j=1,2,\dots ,n,$$

$$\begin{array}{cc}\hfill & vec\left(\widehat{D_1}\right)=\left(\begin{array}{c}vec\left(Re\left(D_1\right)\right)\\ vec\left(Im\left(D_1\right)\right)\end{array}\right),vec\left(\widehat{D}\right)=\left(\begin{array}{c}vec\left(Re\left(D_1\right)\right)\\ vec\left(Im\left(D_1\right)\right)\\ vec\left(Re\left(D_2\right)\right)\\ vec\left(Im\left(D_2\right)\right)\end{array}\right).\hfill \end{array}$$

For a given $D=\left(d_{ij}\right)\in {\mathbb{C}}^{m\times n}$, its Frobenius norm is defined as follows:

$$\begin{array}{cc}\hfill & ∥D_1∥=\sqrt{\sum _{i=1}^m\sum _{j=1}^n{∥d_{ij}∥}^2},{∥d_{ij}∥}^2={\left(\mathrm{Re}{d}_{ij}\right)}^2+{\left(\mathrm{Im}{d}_{ij}\right)}^2.\hfill \end{array}$$

For a given commutative quaternion matrix $D=D_1+D_2\mathbf{j}\in {\mathbb{Q}}_c^{m\times n}$, we have its Frobenius norm, defined as follows:

$$\begin{array}{cc}\hfill & \parallel \widehat{D}\parallel =\sqrt{{∥\mathrm{Re}{D}_1∥}^2+{∥Im{D}_1∥}^2+{∥\mathrm{Re}{D}_2∥}^2+{∥Im{D}_2∥}^2}.\hfill \end{array}$$

It is evident that $∥{\psi }_D∥=\parallel \widehat{D}\parallel =\parallel \mathrm{vec}(\widehat{D})\parallel .$

Definition 3 

([36]). We define the Kronecker product of two matrices, C and D, of size m × n and s × t, respectively, as the (ms) × (nt) matrix:

$$C\otimes D=\left(\begin{array}{cccc}{c}_{11}D& c_{12}D& \cdots & c_{1n}D\\ c_{21}D& c_{22}D& \cdots & c_{2n}D\\ ⋮& ⋮& ⋮& ⋮\\ c_{m1}D& c_{m2}D& \cdots & c_{mn}D\end{array}\right).$$

Definition 4 

([37]). For $G=\left(g_{ij}\right)\in {\mathbb{Q}}_c^{n\times n}$, set

$$\begin{array}{cc}\hfill & g_1=\left(g_{11},\sqrt{2}{g}_{21},\dots ,\sqrt{2}{g}_{n1}\right),g_2=\left(g_{22},\sqrt{2}{g}_{32},\dots ,\sqrt{2}{g}_{n2}\right),\hfill \\ \hfill & \dots ,g_{n-1}=\left(g_{(n-1)(n-1)},\sqrt{2}{g}_{n(n-1)}\right),g_n=g_{nn},\hfill \end{array}$$

and a vector is represented as ${\mathrm{vec}}_S\left(G\right)$, which is constructed as follows:

$${\mathrm{vec}}_S\left(G\right)={\left(g_1,g_2,\dots ,g_{n-1},g_n\right)}^T\in {\mathbb{Q}}_c^{\left(n\right(n+1\left)\right)/2}.$$

Definition 5 

([37]). For $H=\left(h_{ij}\right)\in {\mathbb{Q}}_c^{m\times m}$, set

$$\begin{array}{cc}\hfill & h_1=\left(h_{21},h_{31},\dots ,h_{m1}\right),h_2=\left(h_{32},h_{42},\dots ,h_{m2}\right),\dots ,\hfill \\ \hfill & h_{m-2}=\left(h_{(m-1)(m-2)},h_{m(m-2)}\right),h_{m-1}=h_{m(m-1)},\hfill \end{array}$$

and a vector is represented as $ve{c}_A\left(H\right)$, which is constructed as follows:

$${\mathrm{vec}}_A\left(H\right)=\sqrt{2}{\left(h_1,h_2,\dots ,h_{m-2},h_{m-1}\right)}^T\in {\mathbb{Q}}_c^{\left(m\right(m-1\left)\right)/2}.$$

Proposition 1 

([38]). Assume that $Y\in {\mathbb{R}}^{m\times m}$; then,

$\left(1\right)$

$$Y\in {\mathbb{SR}}^{m\times m}issatisfiedifandonlyif\mathrm{vec}\left(Y\right)=K_S{\mathrm{vec}}_S\left(Y\right),$$

where $K_S\in {\mathbb{R}}^{m^2\times (m(m+1)/2)}$ is as follows:

$$K_S={\displaystyle \frac{1}{\sqrt{2}}}\left(\begin{array}{ccccccccccccc}\sqrt{2}{i}_1& i_2& \cdots & i_{m-1}& i_m& 0& 0& \cdots & 0& \cdots & 0& 0& 0\\ 0& i_1& \cdots & 0& 0& \sqrt{2}{i}_2& i_3& \cdots & i_m& \cdots & 0& 0& 0\\ 0& 0& \cdots & 0& 0& 0& i_2& \cdots & 0& \cdots & 0& 0& 0\\ ⋮& ⋮& & ⋮& ⋮& ⋮& ⋮& & ⋮& & ⋮& ⋮& ⋮\\ 0& 0& \cdots & i_1& 0& 0& 0& \cdots & 0& \cdots & \sqrt{2}{i}_{m-1}& i_m& 0\\ 0& 0& \cdots & 0& i_1& 0& 0& \cdots & i_2& \cdots & 0& i_{m-1}& \sqrt{2}{i}_m\end{array}\right),$$

and $i_j$ denotes the j-th column of an n-dimensional identity matrix.

$\left(2\right)$

$$Y\in {\mathbb{ASR}}^{m\times m}issatisfiedifandonlyif\mathrm{vec}\left(Y\right)=K_A{\mathrm{vec}}_A\left(Y\right),$$

and ${\mathrm{vec}}_A\left(Y\right)$ is described as above and $K_A\in {\mathbb{R}}^{m^2\times (m(m-1)/2)}$ is defined as follows:

$$K_A={\displaystyle \frac{1}{\sqrt{2}}}\left(\begin{array}{ccccccccccc}{i}_2& i_3& \cdots & i_{m-1}& i_m& 0& \cdots & 0& 0& \cdots & 0\\ -i_1& 0& \cdots & 0& 0& i_3& \cdots & i_{m-1}& i_m& \cdots & 0\\ 0& -i_1& \cdots & 0& 0& -i_2& \cdots & 0& 0& \cdots & 0\\ ⋮& ⋮& & ⋮& ⋮& ⋮& & 0& 0& & 0\\ 0& 0& \cdots & -i_1& 0& 0& \cdots & -i_2& 0& \cdots & i_m\\ 0& 0& \cdots & 0& -i_1& 0& \cdots & 0& -i_2& \cdots & -i_{m-1}\end{array}\right),$$

where $i_j$ denotes the j-th column of an n-dimensional identity matrix. Clearly, $K_S^T{K}_S=I_{\left(m\right(m+1\left)\right)/2},K_A^T{K}_A=$$I_{\left(m\right(m-1\left)\right)/2}$.

Lemma 1 

([39]). Suppose that $G=G_1+G_2\mathbf{j}\in \mathbb{H}{\mathbb{Q}}_c^{n\times n}$; then,

$$\left(\begin{array}{c}\mathrm{vec}\left(G_1\right)\\ \mathrm{vec}\left(G_2\right)\end{array}\right)=N\left(\begin{array}{c}{\mathrm{vec}}_S\left(\mathrm{Re}\left(G_1\right)\right)\\ {\mathrm{vec}}_A\left(\mathrm{Im}\left(G_1\right)\right)\\ {\mathrm{vec}}_A\left(\mathrm{Re}\left(G_2\right)\right)\\ {\mathrm{vec}}_A\left(\mathrm{Im}\left(G_2\right)\right)\end{array}\right),$$

in which

$$N=\left(\begin{array}{cccc}{K}_S& \mathbf{i}{K}_A& 0& 0\\ 0& 0& K_A& \mathbf{i}{K}_A\end{array}\right).$$

Lemma 2 

([39]). Let $C=C_1+C_2\mathbf{j}\in {\mathbb{Q}}_c^{m\times n},X=X_1+X_2\mathbf{j}\in \mathbb{H}{\mathbb{Q}}_c^{n\times n}$, and $D=D_1+D_2\mathbf{j}\in$${\mathbb{Q}}_c^{n\times s}$ be given. $C_i\in {\mathbb{C}}^{m\times n},X_i\in {\mathbb{C}}^{n\times n}$, and $D_i\in {\mathbb{C}}^{n\times s}(i=1,2)$. N is the same as defined in Lemma 1. Consequently,

$$\mathrm{vec}\left({\psi }_{CXD}\right)=S\left[\left(D_1^T\otimes C_1+D_2^T\otimes C_2\right)+\left(D_2^T\otimes C_1+D_1^T\otimes C_2\right)\mathbf{j}\right]N\left(\begin{array}{c}{\mathrm{vec}}_S\left(\mathrm{Re}\left(X_1\right)\right)\\ {\mathrm{vec}}_A\left(\mathrm{Im}\left(X_1\right)\right)\\ {\mathrm{vec}}_A\left(\mathrm{Re}\left(X_2\right)\right)\\ {\mathrm{vec}}_A\left(\mathrm{Im}\left(X_2\right)\right)\end{array}\right).$$

Lemma 3 

([40]). Let $=A_1+A_2\mathbf{j}\in {\mathbb{Q}}_c^{m\times n},B=B_1+B_2\mathbf{j}\in {\mathbb{Q}}_c^{m\times n}$ be known, where $A_1,A_2,B_1,B_2\in {\mathbb{C}}^{m\times n}.$ We have the following:

(1). $A=B$ if and only if ${\psi }_A={\psi }_B$;

(2). ${\psi }_{A+B}={\psi }_A+{\psi }_B$ and ${\psi }_{cA}=c{\psi }_A,c\in \mathbb{R}$;

(3). ${\psi }_{AB}={\psi }_{A}S\left(B\right).$

Lemma 4 

([40]). Let $K_t=\left(\begin{array}{cccc}{I}_{nt}& \mathbf{i}{I}_{nt}& 0& 0\\ 0& 0& I_{nt}& \mathbf{i}{I}_{nt}\\ I_{nt}& \mathbf{i}{I}_{nt}& 0& 0\\ 0& 0& I_{nt}& \mathbf{i}{I}_{nt}\end{array}\right)$ be known. Then,

$$\begin{array}{c}\hfill \left(\begin{array}{c}vec\left({\psi }_D\right)\\ vec\left({\psi }_D\right)\end{array}\right)=K_{t}vec\left(\widehat{D}\right),\end{array}$$

where $D=D_1+D_{2}j\in {\mathbb{Q}}_c^{n\times t},D_1,D_2\in {\mathbb{C}}^{n\times t}.$

Lemma 5 

([40]). Let $L=L_1+L_2\mathbf{j}\in {\mathbb{Q}}_c^{m\times n},M=M_1+M_2\mathbf{j}\in {\mathbb{Q}}_c^{n\times s},$ and $N=N_1+N_2\mathbf{j}\in {\mathbb{Q}}_c^{s\times t}$ be known. Then,

$$vec\left({\psi }_{LMN}\right)=(S{\left(N\right)}^T\otimes L_1,S{\left(\mathbf{j}N\right)}^T\otimes L_2)K_{t}vec(\widehat{M}).$$

Lemma 6 

([41]). A solution $x\in$${\mathbb{R}}^n$ exists for the matrix equation $Cx=d$, where $C\in$${\mathbb{R}}^{m\times n}$ and $d\in$${\mathbb{R}}^m$, if and only if

$$C{C}^{†}d=d.$$

Under this condition, the equation’s general solution is captured by the following formula:

$$x=C^{†}d+(I_n-C^{†}C)y,$$

which has an arbitrary matrix y ∈${\mathbb{R}}^n$, and if $rank\left(C\right)=n,$ then it has a solution $C^{†}d$ which is unique.

3. The Solution of System (1) over ${\mathbb{Q}}_c$

This section presents the necessary and sufficient conditions for the commutative quaternion coupled matrix System (1) to have a solution, along with an expression of the general solution to System (1). Let

$$\begin{array}{cc}\hfill & A_1=A_{11}+A_{12}\mathbf{j},A_2=A_{21}+A_{22}\mathbf{j},E_1=E_{11}+E_{12}\mathbf{j},E_2=E_{21}+E_{22}\mathbf{j},G_1=G_{11}+G_{12}\mathbf{j},\hfill \\ \hfill & G_2=G_{21}+G_{22}\mathbf{j}\in {\mathbb{Q}}_c^{s\times n},B_1,B_2,F_1,F_2,H_1,H_2\in {\mathbb{Q}}_c^{t\times m},C_1,C_2,C_3\in {\mathbb{Q}}_c^{s\times m}.\hfill \end{array}$$

Put

$$\begin{array}{cc}\hfill & O_1=ReO,O_2=ImO,P_1=ReP,P_2=ImP,T_1=ReT,\hfill \\ & T_2=ImT,U_1=\left(\begin{array}{c}{O}_1,P_1,T_1\end{array}\right),U_2=\left(\begin{array}{c}{O}_2,P_2,T_2\end{array}\right),\hfill \end{array}$$

$$\begin{array}{cc}\hfill & J=\left(\begin{array}{c}vec\left({\psi }_{C_1}\right)\\ vec\left({\psi }_{C_2}\right)\\ vec\left({\psi }_{C_3}\right)\end{array}\right),J_1=\left(\begin{array}{c}vec\left(Re{\psi }_{C_1}\right)\\ vec\left(Re{\psi }_{C_2}\right)\\ vec\left(Re{\psi }_{C_3}\right)\end{array}\right),\hfill \\ \hfill & J_2=\left(\begin{array}{c}vec\left(Im{\psi }_{C_1}\right)\\ vec\left(Im{\psi }_{C_2}\right)\\ vec\left(Im{\psi }_{C_3}\right)\end{array}\right),\xi =\left(\begin{array}{c}{J}_1\\ J_2\end{array}\right),\hfill \end{array}$$

(6)

where

$$\begin{array}{cc}\hfill & O=\left(\begin{array}{cc}S{\left(B_1\right)}^T\otimes A_{11}& S{\left(\mathbf{j}{B}_1\right)}^T\otimes A_{12}\\ S{\left(F_1\right)}^T\otimes E_{11}& S{\left(\mathbf{j}{F}_1\right)}^T\otimes E_{12}\\ S{\left(H_1\right)}^T\otimes G_{11}& S{\left(\mathbf{j}{H}_1\right)}^T\otimes G_{12}\end{array}\right)K_t,\hfill \\ \hfill & P=\left(\begin{array}{cc}S{\left(B_2\right)}^T\otimes A_{11}& S{\left(\mathbf{j}{B}_2\right)}^T\otimes A_{12}\\ S{\left(F_2\right)}^T\otimes E_{11}& S{\left(\mathbf{j}{F}_2\right)}^T\otimes E_{12}\\ S{\left(H_2\right)}^T\otimes G_{11}& S{\left(\mathbf{j}{H}_2\right)}^T\otimes G_{12}\end{array}\right)K_t,\hfill \\ \hfill & T=\left(\begin{array}{cc}S{\left(B_2\right)}^T\otimes A_{21}& S{\left(\mathbf{j}{B}_2\right)}^T\otimes A_{22}\\ S{\left(F_2\right)}^T\otimes E_{21}& S{\left(\mathbf{j}{F}_2\right)}^T\otimes E_{22}\\ S{\left(H_2\right)}^T\otimes G_{21}& S{\left(\mathbf{j}{H}_2\right)}^T\otimes G_{22}\end{array}\right)K_t.\hfill \end{array}$$

Theorem 1. 

Let $A_1,A_2,E_1,E_2,G_1,G_2\in {\mathbb{Q}}_c^{s\times n},B_1,B_2,F_1,F_2,H_1,H_2\in {\mathbb{Q}}_c^{t\times m}$ and $C_1,C_2,C_3\in {\mathbb{Q}}_c^{s\times m}$ be given, and let $U_1,U_2,\xi$ be defined as in (6). Then, System (1) is solvable with a solution $X_1,X_2,X_3\in {\mathbb{Q}}_c^{n\times t}$, if and only if

$$\begin{array}{c}\hfill \left(\begin{array}{c}{U}_1\\ U_2\end{array}\right){\left(\begin{array}{c}{U}_1\\ U_2\end{array}\right)}^{†}\xi =\xi .\end{array}$$

(7)

In this case, the general solution to System (1) in this context admits the parametric representation:

$$\begin{array}{c}\hfill \Omega =\left\{\begin{array}{c}\left(\begin{array}{ccc}{X}_1& X_2& X_3\end{array}\right)|\left(\begin{array}{c}vec\left(\widehat{X_1}\right)\\ vec\left(\widehat{X_2}\right)\\ vec\left(\widehat{X_3}\right)\end{array}\right)={\left(\begin{array}{c}{U}_1\\ U_2\end{array}\right)}^{†}\xi +\left(\begin{array}{c}{I}_{12n\times t}-{\left(\begin{array}{c}{U}_1\\ U_2\end{array}\right)}^{†}\left(\begin{array}{c}{U}_1\\ U_2\end{array}\right)\end{array}\right)y\end{array}\right\},\end{array}$$

(8)

where y is an arbitrary vector with an appropriate order. The system (1) possesses a unique solution $\left(\begin{array}{ccc}{X}_1& X_2& X_3\end{array}\right)\in \Omega$, if and only if the following conditions hold:

$$\begin{array}{c}\hfill rank\left(\begin{array}{c}{U}_1\\ U_2\end{array}\right)=12n\times t.\end{array}$$

(9)

In this case, the unique solution to System (1) is as follows:

$$\begin{array}{c}\hfill \Omega =\left\{\begin{array}{c}\left(\begin{array}{ccc}{X}_1& X_2& X_3\end{array}\right)|\left(\begin{array}{c}vec\left(\widehat{X_1}\right)\\ vec\left(\widehat{X_2}\right)\\ vec\left(\widehat{X_3}\right)\end{array}\right)={\left(\begin{array}{c}{U}_1\\ U_2\end{array}\right)}^{†}\xi \end{array}\right\}.\end{array}$$

(10)

Proof. 

Based on Lemmas 3 and 5, we have the following:

$$\begin{array}{cc}\hfill \left(1\right)& \iff \left\{\begin{array}{c}{\psi }_{A_1{X}_1{B}_1}+{\psi }_{A_1{X}_2{B}_2}+{\psi }_{A_2{X}_3{B}_2}={\psi }_{C_1},\\ {\psi }_{E_1{X}_1{F}_1}+{\psi }_{E_1{X}_2{F}_2}+{\psi }_{E_2{X}_3{F}_2}={\psi }_{C_2},\\ {\psi }_{G_1{X}_1{H}_1}+{\psi }_{G_1{X}_2{H}_2}+{\psi }_{G_2{X}_3{H}_2}={\psi }_{C_3}\end{array}\right.\hfill \\ \hfill & \iff Ovec\left(\widehat{X_1}\right)+Pvec\left(\widehat{X_2}\right)+Tvec\left(\widehat{X_3}\right)=J\hfill \\ \hfill & \iff (ReO+\mathbf{i}ImO)vec\left(\widehat{X_1}\right)+(ReP+\mathbf{i}ImP)vec\left(\widehat{X_2}\right)\hfill \\ \hfill & \hspace{1em}+(ReT+\mathbf{i}ImT)vec\left(\widehat{X_3}\right)=J_1+\mathbf{i}{J}_2\hfill \\ \hfill & \iff \left(\begin{array}{ccc}ReO& ReP& ReT\\ ImO& ImP& ImT\end{array}\right)\left(\begin{array}{c}vec\left(\widehat{X_1}\right)\\ vec\left(\widehat{X_2}\right)\\ vec\left(\widehat{X_3}\right)\end{array}\right)=\xi \hfill \\ \hfill & \iff \left(\begin{array}{c}{U}_1\\ U_2\end{array}\right)\left(\begin{array}{c}vec\left(\widehat{X_1}\right)\\ vec\left(\widehat{X_2}\right)\\ vec\left(\widehat{X_3}\right)\end{array}\right)=\xi .\hfill \end{array}$$

(11)

According to Lemma 6, the existence of a solution

$$\left(\begin{array}{ccc}{X}_1& X_2& X_3\end{array}\right)\in \Omega$$

for System (11) is guaranteed if and only if (7) holds, i.e., the system of matrix equations (1) has a solution

$$\left(\begin{array}{ccc}{X}_1& X_2& X_3\end{array}\right)\in \Omega ,$$

if and only if (7) holds. When Equation (7) is satisfied, the general solution for System (1) can be formulated as follows:

$$\left(\begin{array}{c}vec\left(\widehat{X_1}\right)\\ vec\left(\widehat{X_2}\right)\\ vec\left(\widehat{X_3}\right)\end{array}\right)={\left(\begin{array}{c}{U}_1\\ U_2\end{array}\right)}^{†}\xi +\left(\begin{array}{c}{I}_{12n\times t}-{\left(\begin{array}{c}{U}_1\\ U_2\end{array}\right)}^{†}\left(\begin{array}{c}{U}_1\\ U_2\end{array}\right)\end{array}\right)y,$$

i.e., (8) holds. Moreover, based on Lemma 6, the matrix equation (11) has a unique solution

$$\left(\begin{array}{ccc}{X}_1& X_2& X_3\end{array}\right)\in \Omega ,$$

if and only if the following condition holds:

$$r\left(\begin{array}{c}{U}_1\\ U_2\end{array}\right)=12n\times t.$$

In this case, the solution to the matrix equation is (10), i.e., the system of matrix equations (1) has an unique solution

$$\left(\begin{array}{ccc}{X}_1& X_2& X_3\end{array}\right)\in \Omega ,$$

if and only if (9) holds, and the unique solution to the matrix equations (1) is as follows:

$$\begin{array}{c}\hfill \left(\begin{array}{c}vec\left(\widehat{X_1}\right)\\ vec\left(\widehat{X_2}\right)\\ vec\left(\widehat{X_3}\right)\end{array}\right)={\left(\begin{array}{c}{U}_1\\ U_2\end{array}\right)}^{†}\xi .\end{array}$$

   □

Subsequently, the Moore–Penrose generalized inverse of the column block matrix is investigated. Let

$$\begin{array}{cc}\hfill & p=6m\times s,\hfill \\ \hfill & \widehat{G}=\left(I_{12n\times t}-U_1^{†}{U}_1\right)U_2^T,\hfill \\ \hfill & \widehat{J}=(I_p+(I_p-{\widehat{G}}^{†}\widehat{G})U_2{U}_1^{†}{U}_1^{†T}{U}_2^T{(I_p-{\widehat{G}}^{†}\widehat{G}))}^{-1},\hfill \\ \hfill & \widehat{R}={\widehat{G}}^{†}+\left(I_p-{\widehat{G}}^{†}\widehat{G}\right)\widehat{J}{U}_2{U}_1^{†}{U}_1^{†T}\left(I_{12n\times t}-U_2^T{\widehat{G}}^{†}\right),\hfill \\ \hfill & {\widehat{Q}}_1=I_p-U_1{U}_1^{†}-U_1^{†T}{U}_2^T\widehat{J}\left(I_p-{\widehat{G}}^{†}\widehat{G}\right)U_2{U}_1^{†},\hfill \\ \hfill & {\widehat{Q}}_2=-U_1^{†T}{U}_2^T\left(I_p-{\widehat{G}}^{†}\widehat{G}\right)\widehat{J},\hfill \\ \hfill & {\widehat{Q}}_3=\left(I_p-{\widehat{G}}^{†}\widehat{G}\right)\widehat{J}.\hfill \end{array}$$

According to the results of [42], we have the following:

$$\begin{array}{cc}\hfill & {\left(\begin{array}{c}{U}_1\\ U_2\end{array}\right)}^{†}=\left(\begin{array}{cc}{U}_1^{†}-{\widehat{R}}_1^T{U}_2{U}_1^{†}& {\widehat{R}}_1^T\end{array}\right),{\left(\begin{array}{c}{U}_1\\ U_2\end{array}\right)}^{†}\left(\begin{array}{c}{U}_1\\ U_2\end{array}\right)=U_1^{†}{U}_1+\widehat{G}{\widehat{G}}^{†}.\hfill \\ \hfill & I_{2p}-\left(\begin{array}{c}{U}_1\\ U_2\end{array}\right){\left(\begin{array}{c}{U}_1\\ U_2\end{array}\right)}^{†}=\left(\begin{array}{cc}{\widehat{Q}}_1& {\widehat{Q}}_2\\ {\widehat{Q}}_2^T& {\widehat{Q}}_3\end{array}\right).\hfill \end{array}$$

Corollary 1. 

System (1) has a solution $\left(\begin{array}{ccc}{X}_1& X_2& X_3\end{array}\right)$, if and only if

$$\left(\begin{array}{cc}{\widehat{Q}}_1& {\widehat{Q}}_2\\ {\widehat{Q}}_2^T& {\widehat{Q}}_3\end{array}\right)\xi =0.$$

(12)

In this circumstance, the set of general solutions to (1) can be expressed as follows:

$$\Omega =\left\{\left(\begin{array}{ccc}{X}_1& X_2& X_3\end{array}\right)|\left(\begin{array}{c}\mathrm{vec}\left(\widehat{X_1}\right)\\ \mathrm{vec}\left(\widehat{X_2}\right)\\ \mathrm{vec}\left(\widehat{X_3}\right)\end{array}\right)=\left(\begin{array}{cc}{U}_1^T-{\widehat{R}}^T{U}_2{U}_1^{†}& {\widehat{R}}^T\end{array}\right)\xi +(I_{12n\times t}-U_1^{†}{U}_1-\widehat{G}{\widehat{G}}^{†})y\right\},$$

(13)

where y is an arbitrary vector with an appropriate order. Moreover, when (12) holds, the system of commutative quaternion matrix equations (1) has a unique solution $\left[\begin{array}{ccc}X& Y& Z\end{array}\right]\in \Omega$ if and only if (9) holds. In this circumstance,

$$\Omega =\left\{\left(\begin{array}{ccc}{X}_1& X_2& X_3\end{array}\right)|\left(\begin{array}{c}\mathrm{vec}\left(\widehat{X_1}\right)\\ \mathrm{vec}\left(\widehat{X_2}\right)\\ \mathrm{vec}\left(\widehat{X_3}\right)\end{array}\right)=\left(\begin{array}{cc}{U}_1^T-{\widehat{R}}^T{U}_2{U}_1^{†}& {\widehat{R}}^T\end{array}\right)\xi \right\}.$$

(14)

Corollary 2. 

If the condition in Corollary 3.2 is satisfied, 1 is satisfied. Then, the optimization problem

$$\underset{(X_{!},X_2,X_3)\in \Omega }{min}\left({\parallel {\psi }_{X_1}\parallel }^2+{\parallel {\psi }_{X_2}\parallel }^2+{\parallel {\psi }_{X_3}\parallel }^2\right)$$

has a unique minimizer $\left(\begin{array}{ccc}{X}_{1\eta }& X_{2\eta }& X_{3\eta }\end{array}\right)$ and satisfies the following:

$$\left(\begin{array}{c}\mathrm{vec}\left({\widehat{X}}_{1\eta }\right)\\ \mathrm{vec}\left({\widehat{X}}_{2\eta }\right)\\ \mathrm{vec}\left({\widehat{X}}_{3\eta }\right)\end{array}\right)=\left(\begin{array}{cc}{U}_1^T-{\widehat{R}}^T{U}_2{U}_1^{†}& {\widehat{R}}^T\end{array}\right)\xi .$$

(15)

Proof. 

Based on (14), we can see that the solution set Ω is a nonempty closed convex set. Therefore,

$$\begin{array}{cc}\hfill & \underset{(X_1,X_2,X_3)\in \Omega }{min}\left({\parallel \psi X_1\parallel }^2+{\parallel {\psi }_{X_2}\parallel }^2+{\parallel {\psi }_{X_3}\parallel }^2\right)\hfill \\ \hfill & =\underset{(X_1,X_2,X_3)\in \Omega }{min}\left({\parallel \widehat{X_1}\parallel }^2+{\parallel \widehat{X_2}\parallel }^2+{\parallel \widehat{X_3}\parallel }^2\right)\hfill \\ \hfill & =\underset{(X_1,X_2,X_3)\in \Omega }{min}\left(\parallel \mathrm{vec}\left(\widehat{X_1}\right){\parallel }^2+\parallel \mathrm{vec}\left(\widehat{X_2}\right){\parallel }^2+{\parallel \mathrm{vec}\left(\widehat{X_3}\right)\parallel }^2\right)\hfill \\ \hfill & =\underset{(X_1,X_2,X_3)\in \Omega }{min}∥\left(\begin{array}{c}\mathrm{vec}\left(\widehat{X_1}\right)\\ \mathrm{vec}\left(\widehat{X_2}\right)\\ \mathrm{vec}\left(\widehat{X_3}\right)\end{array}\right)∥.\hfill \end{array}$$

Based on Corollary 1, we have that $\left(\begin{array}{c}\mathrm{vec}\left(\widehat{X_{!}}\right)\\ \mathrm{vec}\left(\widehat{X_2}\right)\\ \mathrm{vec}\left(\widehat{X_3}\right)\end{array}\right)$ is presented as (15).    □

4. The Hermitian Solution of System (1) over ${\mathbb{Q}}_c$ 

This section establishes the necessary and sufficient conditions for the existence of a Hermitian solution to System (1) over the commutative quaternion ring and provides an explicit formula for such a solution. Let $A_1=A_{11}+A_{12}\mathbf{j}$,  $A_2=A_{21}+A_{22}\mathbf{j}$,  $E_1=E_{11}+E_{12}\mathbf{j}$,  $E_2=E_{21}+E_{22}\mathbf{j}$,  $G_1=G_{11}+G_{12}\mathbf{j}$,  $G_2=G_{21}+G_{22}\mathbf{j}\in {\mathbb{Q}}_c^{s\times n}$,  $B_1$,  $B_2$,  $F_1$,  $F_2$,  $H_1$,  $H_2\in {\mathbb{Q}}_c^{n\times m}$ and $C_1,C_2,C_3\in {\mathbb{Q}}_c^{s\times m}$. Put

$$\begin{array}{cc}\hfill & O_1=ReO,O_2=ImO,P_1=ReP,\hfill \\ & P_2=ImP,T_1=ReT,T_2=ImT,\hfill \\ & U_1=\left(\begin{array}{c}{O}_1,P_1,T_1\end{array}\right),U_2=\left(\begin{array}{c}{O}_2,P_2,T_2\end{array}\right),\hfill \\ & J=\left(\begin{array}{c}vec\left({\psi }_{C_1}\right)\\ vec\left({\psi }_{C_2}\right)\\ vec\left({\psi }_{C_3}\right)\end{array}\right),J_1=\left(\begin{array}{c}vec\left(Re{\psi }_{C_1}\right)\\ vec\left(Re{\psi }_{C_2}\right)\\ vec\left(Re{\psi }_{C_3}\right)\end{array}\right),\hfill \\ \hfill & J_2=\left(\begin{array}{c}vec\left(Im{\psi }_{C_1}\right)\\ vec\left(Im{\psi }_{C_2}\right)\\ vec\left(Im{\psi }_{C_3}\right)\end{array}\right),\xi =\left(\begin{array}{c}{J}_1\\ J_2\end{array}\right),\hfill \\ \hfill & \mathbb{V}=\left(\begin{array}{cccc}{K}_S& 0& 0& 0\\ 0& K_A& 0& 0\\ 0& 0& K_A& 0\\ 0& 0& 0& K_A\end{array}\right),\hfill \\ \hfill & \mathbb{W}=\left(\begin{array}{ccc}\mathbb{V}& 0& 0\\ 0& \mathbb{V}& 0\\ 0& 0& \mathbb{V}\end{array}\right),vec\left({\mathbb{X}}_1\right)=\left(\begin{array}{c}ve{c}_S\left(Re\left(X_{11}\right)\right)\\ ve{c}_A\left(Im\left(X_{11}\right)\right)\\ ve{c}_A\left(Re\left(X_{12}\right)\right)\\ ve{c}_A\left(Im\left(X_{12}\right)\right)\end{array}\right),\hfill \end{array}$$

(16)

where

$$\begin{array}{cc}\hfill & vec\left({\mathbb{X}}_2\right)=\left(\begin{array}{c}ve{c}_S\left(Re\left(X_{21}\right)\right)\\ ve{c}_A\left(Im\left(X_{21}\right)\right)\\ ve{c}_A\left(Re\left(X_{22}\right)\right)\\ ve{c}_A\left(Im\left(X_{22}\right)\right)\end{array}\right),\hfill \\ \hfill & vec\left({\mathbb{X}}_3\right)=\left(\begin{array}{c}ve{c}_S\left(Re\left(X_{31}\right)\right)\\ ve{c}_A\left(Im\left(X_{31}\right)\right)\\ ve{c}_A\left(Re\left(X_{32}\right)\right)\\ ve{c}_A\left(Im\left(X_{32}\right)\right)\end{array}\right),\hfill \\ \hfill & O=\left(\begin{array}{c}S[(B_{11}^T\otimes A_{11}+B_{12}^T\otimes A_{12})+(B_{12}^T\otimes A_{11}+B_{11}^T\otimes A_{12})\mathbf{j}]\\ S[(F_{11}^T\otimes E_{11}+F_{12}^T\otimes E_{12})+(F_{12}^T\otimes E_{11}+F_{11}^T\otimes E_{12})\mathbf{j}]\\ S[(H_{11}^T\otimes G_{11}+H_{12}^T\otimes G_{12})+(H_{12}^T\otimes G_{11}+H_{11}^T\otimes G_{12})\mathbf{j}]\end{array}\right)N,\hfill \\ \hfill & P=\left(\begin{array}{c}S[(B_{21}^T\otimes A_{11}+B_{22}^T\otimes A_{12})+(B_{22}^T\otimes A_{11}+B_{21}^T\otimes A_{12})\mathbf{j}]\\ S[(F_{21}^T\otimes E_{11}+F_{22}^T\otimes E_{12})+(F_{22}^T\otimes E_{11}+F_{21}^T\otimes E_{12})\mathbf{j}]\\ S[(H_{21}^T\otimes G_{11}+H_{22}^T\otimes G_{12})+(H_{22}^T\otimes G_{11}+H_{21}^T\otimes G_{12})\mathbf{j}]\end{array}\right)N,\hfill \\ \hfill & T=\left(\begin{array}{c}S[(B_{21}^T\otimes A_{21}+B_{22}^T\otimes A_{22})+(B_{22}^T\otimes A_{21}+B_{21}^T\otimes A_{22})\mathbf{j}]\\ S[(F_{21}^T\otimes E_{21}+F_{22}^T\otimes E_{22})+(F_{22}^T\otimes E_{21}+F_{21}^T\otimes E_{22})\mathbf{j}]\\ S[(H_{21}^T\otimes G_{21}+H_{22}^T\otimes G_{22})+(H_{22}^T\otimes G_{21}+H_{21}^T\otimes G_{22})\mathbf{j}]\end{array}\right)N.\hfill \end{array}$$

Theorem 2. 

Let $A_1,A_2,E_1,E_2,G_1,G_2\in {\mathbb{Q}}_c^{s\times n},B_1,B_2,F_1,F_2,H_1,H_2\in {\mathbb{Q}}_c^{n\times m}$ and $C_1,C_2,C_3\in {\mathbb{Q}}_c^{s\times m}$ be given, and let $U_1,U_2,\xi$ be defined as in (16). Then, System (1) has a solution $X_1=X_{11}+X_{12}\mathbf{j},X_2=X_{21}+X_{22}\mathbf{j},X_3=X_{31}+X_{32}\mathbf{j}\in \mathbb{H}{\mathbb{Q}}_c^{n\times n}$, if and only if

$$\begin{array}{c}\hfill \left(\begin{array}{c}{U}_1\\ U_2\end{array}\right){\left(\begin{array}{c}{U}_1\\ U_2\end{array}\right)}^{†}\xi =\xi .\end{array}$$

(17)

Under this condition, the Hermitian solution can be expressed as follows:

$$\begin{array}{c}\hfill \Omega =\left\{\begin{array}{c}\left(\begin{array}{ccc}{X}_1& X_2& X_3\end{array}\right)|\left(\begin{array}{c}vec\left(\widehat{X_1}\right)\\ vec\left(\widehat{X_2}\right)\\ vec\left(\widehat{X_3}\right)\end{array}\right)=\mathbb{W}{\left(\begin{array}{c}{U}_1\\ U_2\end{array}\right)}^{†}\xi +\mathbb{W}\left(\begin{array}{c}{I}_{6n^2-3n}-{\left(\begin{array}{c}{U}_1\\ U_2\end{array}\right)}^{†}\left(\begin{array}{c}{U}_1\\ U_2\end{array}\right)\end{array}\right)y\end{array}\right\},\end{array}$$

(18)

where y is any vector with the proper size. Additionally, a unique solution $\left(\begin{array}{ccc}{X}_1& X_2& X_3\end{array}\right)$ $\in \Omega$ is possessed by System (1) if and only if

$$\begin{array}{c}\hfill rank\left(\begin{array}{c}{U}_1\\ U_2\end{array}\right)=6n^2-3n.\end{array}$$

(19)

In this instance,

$$\begin{array}{c}\hfill \Omega =\left\{\begin{array}{c}\left(\begin{array}{ccc}{X}_1& X_2& X_3\end{array}\right)|\left(\begin{array}{c}vec\left(\widehat{X_1}\right)\\ vec\left(\widehat{X_2}\right)\\ vec\left(\widehat{X_3}\right)\end{array}\right)=\mathbb{W}{\left(\begin{array}{c}{U}_1\\ U_2\end{array}\right)}^{†}\xi \end{array}\right\}.\end{array}$$

(20)

Proof. 

According to Lemmas 1–3,

$$\begin{array}{cc}\hfill \left(1\right)& \iff \left\{\begin{array}{c}{\psi }_{A_1{X}_1{B}_1}+{\psi }_{A_1{X}_2{B}_2}+{\psi }_{A_2{X}_3{B}_2}={\psi }_{C_1},\\ {\psi }_{E_1{X}_1{F}_1}+{\psi }_{E_1{X}_2{F}_2}+{\psi }_{E_2{X}_3{F}_2}={\psi }_{C_2},\\ {\psi }_{G_1{X}_1{H}_1}+{\psi }_{G_1{X}_2{H}_2}+{\psi }_{G_2{X}_3{H}_2}={\psi }_{C_3}\end{array}\right.\hfill \\ \hfill & \iff O\left(\begin{array}{c}ve{c}_S\left(Re\left(X_{11}\right)\right)\\ ve{c}_A\left(Im\left(X_{11}\right)\right)\\ ve{c}_A\left(Re\left(X_{12}\right)\right)\\ ve{c}_A\left(Im\left(X_{12}\right)\right)\end{array}\right)+P\left(\begin{array}{c}ve{c}_S\left(Re\left(X_{21}\right)\right)\\ ve{c}_A\left(Im\left(X_{21}\right)\right)\\ ve{c}_A\left(Re\left(X_{22}\right)\right)\\ ve{c}_A\left(Im\left(X_{22}\right)\right)\end{array}\right)+T\left(\begin{array}{c}ve{c}_S\left(Re\left(X_{31}\right)\right)\\ ve{c}_A\left(Im\left(X_{31}\right)\right)\\ ve{c}_A\left(Re\left(X_{32}\right)\right)\\ ve{c}_A\left(Im\left(X_{32}\right)\right)\end{array}\right)=J\hfill \\ \hfill & \iff (ReO+\mathbf{i}ImO)\left(\begin{array}{c}ve{c}_S\left(Re\left(X_{11}\right)\right)\\ ve{c}_A\left(Im\left(X_{11}\right)\right)\\ ve{c}_A\left(Re\left(X_{12}\right)\right)\\ ve{c}_A\left(Im\left(X_{12}\right)\right)\end{array}\right)+(ReP+\mathbf{i}ImP)\left(\begin{array}{c}ve{c}_S\left(Re\left(X_{21}\right)\right)\\ ve{c}_A\left(Im\left(X_{21}\right)\right)\\ ve{c}_A\left(Re\left(X_{22}\right)\right)\\ ve{c}_A\left(Im\left(X_{22}\right)\right)\end{array}\right)\hfill \\ \hfill & \hspace{1em}+(ReT+\mathbf{i}ImT)\left(\begin{array}{c}ve{c}_S\left(Re\left(X_{31}\right)\right)\\ ve{c}_A\left(Im\left(X_{31}\right)\right)\\ ve{c}_A\left(Re\left(X_{32}\right)\right)\\ ve{c}_A\left(Im\left(X_{32}\right)\right)\end{array}\right)=J_1+\mathbf{i}{J}_2\hfill \end{array}$$

$$\begin{array}{cc}\hfill & \iff \left(\begin{array}{ccc}ReO& ReP& ReT\\ ImO& ImP& ImT\end{array}\right)\left(\begin{array}{c}vec\left({\mathbb{X}}_1\right)\\ vec\left({\mathbb{X}}_2\right)\\ vec\left({\mathbb{X}}_3\right)\end{array}\right)=\xi \hfill \\ \hfill & \iff \left(\begin{array}{c}{U}_1\\ U_2\end{array}\right)\left(\begin{array}{c}vec\left({\mathbb{X}}_1\right)\\ vec\left({\mathbb{X}}_2\right)\\ vec\left({\mathbb{X}}_3\right)\end{array}\right)=\xi .\hfill \end{array}$$

(21)

Lemma 6 establishes that the coupled system (21) allows for a solution triplet $\left(\begin{array}{ccc}{X}_1& X_2& X_3\end{array}\right)\in \Omega$ if and only if the compatibility condition (17) holds, and

$$\begin{array}{cc}\hfill & \left(\begin{array}{c}vec\left({\mathbb{X}}_1\right)\\ vec\left({\mathbb{X}}_2\right)\\ vec\left({\mathbb{X}}_3\right)\end{array}\right)={\left(\begin{array}{c}{U}_1\\ U_2\end{array}\right)}^{†}\xi +\left(\begin{array}{c}{I}_{6n^2-3n}-{\left(\begin{array}{c}{U}_1\\ U_2\end{array}\right)}^{†}\left(\begin{array}{c}{U}_1\\ U_2\end{array}\right)\end{array}\right)y,\hfill \\ \hfill & \widehat{X_1}=\left(\begin{array}{c}Re\left(X_{11}\right)\\ Im\left(X_{11}\right)\\ Re\left(X_{12}\right)\\ Im\left(X_{12}\right)\end{array}\right),\widehat{X_2}=\left(\begin{array}{c}Re\left(X_{21}\right)\\ Im\left(X_{21}\right)\\ Re\left(X_{22}\right)\\ Im\left(X_{22}\right)\end{array}\right),\widehat{X_3}=\left(\begin{array}{c}Re\left(X_{31}\right)\\ Im\left(X_{31}\right)\\ Re\left(X_{32}\right)\\ Im\left(X_{32}\right)\end{array}\right),\hfill \\ \hfill & vec\left(\widehat{X_1}\right)=\left(\begin{array}{c}vec\left(Re\left(X_{11}\right)\right)\\ vec\left(Im\left(X_{11}\right)\right)\\ vec\left(Re\left(X_{12}\right)\right)\\ vec\left(Im\left(X_{12}\right)\right)\end{array}\right)=\left(\begin{array}{cccc}{K}_S& 0& 0& 0\\ 0& K_A& 0& 0\\ 0& 0& K_A& 0\\ 0& 0& 0& K_A\end{array}\right)\left(\begin{array}{c}ve{c}_S\left(Re\left(X_{11}\right)\right)\\ ve{c}_A\left(Im\left(X_{11}\right)\right)\\ ve{c}_A\left(Re\left(X_{12}\right)\right)\\ ve{c}_A\left(Im\left(X_{12}\right)\right)\end{array}\right)=\mathbb{V}vec\left({\mathbb{X}}_1\right).\hfill \end{array}$$

Similarly, it follows that

$$\begin{array}{c}\hfill vec\left(\widehat{X_2}\right)=\mathbb{V}vec\left({\mathbb{X}}_2\right),vec\left(\widehat{X_3}\right)=\mathbb{V}vec\left({\mathbb{X}}_3\right).\end{array}$$

Thus, we have the following:

$$\begin{array}{cc}\hfill & \left(\begin{array}{c}vec\left(\widehat{X_1}\right)\\ vec\left(\widehat{X_2}\right)\\ vec\left(\widehat{X_3}\right)\end{array}\right)=\left(\begin{array}{ccc}\mathbb{V}& 0& 0\\ 0& \mathbb{V}& 0\\ 0& 0& \mathbb{V}\end{array}\right)\left(\begin{array}{c}vec\left({\mathbb{X}}_1\right)\\ vec\left({\mathbb{X}}_2\right)\\ vec\left({\mathbb{X}}_3\right)\end{array}\right)=\mathbb{W}\left(\begin{array}{c}vec\left({\mathbb{X}}_1\right)\\ vec\left({\mathbb{X}}_2\right)\\ vec\left({\mathbb{X}}_3\right)\end{array}\right)\hfill \\ \hfill & =\mathbb{W}{\left(\begin{array}{c}{U}_1\\ U_2\end{array}\right)}^{†}\xi +\mathbb{W}\left(\begin{array}{c}{I}_{6n^2-3n}-{\left(\begin{array}{c}{U}_1\\ U_2\end{array}\right)}^{†}\left(\begin{array}{c}{U}_1\\ U_2\end{array}\right)\end{array}\right)y.\hfill \end{array}$$

Consequently, the existence of a Hermitian solution $\left(\begin{array}{ccc}{X}_1& X_2& X_3\end{array}\right)\in \Omega$ for System (1) is contingent upon the joint satisfaction of conditions (17) and (18). Furthermore, assuming that (17) is true, System (1) possesses a unique solution $\left(\begin{array}{ccc}{X}_1& X_2& X_3\end{array}\right)\in \Omega$ if and only if

$${\left(\begin{array}{c}{U}_1\\ U_2\end{array}\right)}^{†}\left(\begin{array}{c}{U}_1\\ U_2\end{array}\right)=I_{6n^2-3n},$$

which means that (19) holds, and that (20) clearly holds.    □

5. An Application of System (1) over ${\mathbb{Q}}_c$

This section determines the necessary and sufficient conditions for the reducible solution of the classical matrix equations System (2) over the commutative quaternion ring.

Theorem 3. 

Let $A,E,G\in {\mathbb{Q}}_c^{s\times n},K,K^{-1}\in {\mathbb{Q}}_c^{n\times n},B,F$,

$$H\in {\mathbb{Q}}_c^{n\times m},A_1,G_1,E_1\in {\mathbb{Q}}_c^{s\times (n-k)},B_1,F_1,H_1\in {\mathbb{Q}}_c^{(n-k)\times m},$$

$A_2,G_2,E_2\in {\mathbb{Q}}_c^{s\times k},B_2,F_2,H_2\in {\mathbb{Q}}_c^{k\times m}$ be given. Put

$$\begin{array}{c}\hfill AK=\left(\begin{array}{cc}{A}_1& A_2\end{array}\right),K^{-1}B=\left(\begin{array}{c}{B}_1\\ B_2\end{array}\right),\\ \hfill EK=\left(\begin{array}{cc}{E}_1& E_2\end{array}\right),K^{-1}F=\left(\begin{array}{c}{F}_1\\ F_2\end{array}\right),\\ \hfill GK=\left(\begin{array}{cc}{G}_1& G_2\end{array}\right),K^{-1}H=\left(\begin{array}{c}{H}_1\\ H_2\end{array}\right).\end{array}$$

(22)

The subsequent statements are equivalent.

(1) System (2) possesses a solution $X\in {\mathbb{Q}}_c^{n\times n}$, which is reducible.

(2)

$$\begin{array}{c}\hfill \left(\begin{array}{c}{U}_1\\ U_2\end{array}\right){\left(\begin{array}{c}{U}_1\\ U_2\end{array}\right)}^{†}\xi =\xi .\end{array}$$

(23)

If (23) holds, a k-reducible solution X of System (2) concerning K can be articulated by the following:

$$\begin{array}{c}\hfill X=K\left(\begin{array}{cc}{X}_1& X_2\\ 0& X_3\end{array}\right)K^{-1},\end{array}$$

(24)

where

$$\begin{array}{c}\hfill \Omega =\left\{\begin{array}{c}\left(\begin{array}{ccc}{X}_1& X_2& X_3\end{array}\right)|\left(\begin{array}{c}vec\left(\widehat{X_1}\right)\\ vec\left(\widehat{X_2}\right)\\ vec\left(\widehat{X_3}\right)\end{array}\right)={\left(\begin{array}{c}{U}_1\\ U_2\end{array}\right)}^{†}\xi +\left(\begin{array}{c}{I}_{4{(n-k)}^2}-{\left(\begin{array}{c}{U}_1\\ U_2\end{array}\right)}^{†}\left(\begin{array}{c}{U}_1\\ U_2\end{array}\right)\end{array}\right)y\end{array}\right\},\end{array}$$

and y is an arbitrary vector which has a suitable order, $X_1\in {\mathbb{Q}}_c^{(n-k)\times (n-k)}$, $X_3\in {\mathbb{Q}}_c^{k\times k}$.

Proof. 

(1) ⇒ (2): We suppose that $X\in {\mathbb{Q}}_c^{n\times n}$ is the reducible solution of System (2); then, X possesses the structure of (24). According to (22), we have the following:

$$\left\{\begin{array}{c}\hfill \left(\begin{array}{cc}{A}_1& A_2\end{array}\right)\left(\begin{array}{cc}{X}_1& X_2\\ 0& X_3\end{array}\right)\left(\begin{array}{c}{B}_1\\ B_2\end{array}\right)=C_1,\\ \hfill \left(\begin{array}{cc}{E}_1& E_2\end{array}\right)\left(\begin{array}{cc}{X}_1& X_2\\ 0& X_3\end{array}\right)\left(\begin{array}{c}{F}_1\\ F_2\end{array}\right)=C_2,\\ \hfill \left(\begin{array}{cc}{G}_1& G_2\end{array}\right)\left(\begin{array}{cc}{X}_1& X_2\\ 0& X_3\end{array}\right)\left(\begin{array}{c}{H}_1\\ H_2\end{array}\right)=C_3,\end{array}\right.$$

(25)

i.e.,

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{A}_1{X}_1{B}_1+A_1{X}_2{B}_2+A_2{X}_3{B}_2=C_1,\\ E_1{X}_1{F}_1+E_1{X}_2{F}_2+E_2{X}_3{F}_2=C_2,\\ G_1{X}_1{H}_1+G_1{X}_2{H}_2+G_2{X}_3{H}_2=C_3.\end{array}\right.\end{array}$$

Hence, $(X_1,X_2,X_3)$ form a solution to System (1). According to Theorem 1, if (19) is true, then $X_1,X_2,X_3\in \Omega .$ Substituting $X_1,X_2,$ and $X_3$ into (24) leads to the conclusion that (2) holds.

(2) ⇒ (1): We assume that (2) holds. Based on Theorem 1, System (1) is solvable with the solution $X_1,X_2,X_3\in \Omega$. By way of (22) and (25), it is clear that $X_1\in {\mathbb{Q}}_c^{(n-k)\times (n-k)}, X_3\in {\mathbb{Q}}_c^{k\times k}$, and that X with the structure of (24) is a k-reducible solution to System (2).    □

6. Numerical Illustration of an Algorithmic Approach

This section provides an algorithm, as well as two examples to demonstrate Theorem 1.

If System (1) is solvable, then we have that

$$\begin{array}{c}\hfill {\theta }_1=\parallel \left(\begin{array}{c}{U}_1\\ U_2\end{array}\right){\left(\begin{array}{c}{U}_1\\ U_2\end{array}\right)}^{†}\xi -\xi \parallel ,{\theta }_2=\parallel \left(\begin{array}{cc}{\widehat{Q}}_1& {\widehat{Q}}_2\\ {\widehat{Q}}_2^T& {\widehat{Q}}_3\end{array}\right)\xi \parallel \end{array}$$

and

$$\begin{array}{c}\hfill {\theta }_3=\parallel I-\left(\begin{array}{c}{U}_1\\ U_2\end{array}\right){\left(\begin{array}{c}{U}_1\\ U_2\end{array}\right)}^{†}-\left(\begin{array}{cc}{\widehat{Q}}_1& {\widehat{Q}}_2\\ {\widehat{Q}}_2^T& {\widehat{Q}}_3\end{array}\right)\parallel \end{array}$$

are small enough.

Example 1. 

Let

$$\begin{array}{cc}\hfill & A_1=\left(\begin{array}{cc}\mathbf{i}& \mathbf{j}\\ 0& \mathbf{i}\end{array}\right),A_2=\left(\begin{array}{cc}1+\mathbf{i}& 0.5\mathbf{j}\\ 0.5\mathbf{j}& 1-\mathbf{i}\end{array}\right),E_1=\left(\begin{array}{cc}\mathbf{j}& \mathbf{i}\\ 0& 0\end{array}\right),\hfill \\ & E_2=\left(\begin{array}{cc}\mathbf{k}& \mathbf{j}\\ \mathbf{i}& \mathbf{i}\end{array}\right),G_1=\left(\begin{array}{cc}\mathbf{j}& 0\\ 0& \mathbf{j}\end{array}\right),G_2=\left(\begin{array}{cc}\mathbf{i}& \mathbf{k}\\ \mathbf{j}& \mathbf{j}\end{array}\right),\hfill \\ \hfill & B_1=\left(\begin{array}{cc}\mathbf{j}& \mathbf{k}\\ \mathbf{i}& 0\end{array}\right),B_2=\left(\begin{array}{cc}\mathbf{i}& \mathbf{j}\\ \mathbf{k}& \mathbf{i}\end{array}\right),F_1=\left(\begin{array}{cc}1+\mathbf{i}& 1+\mathbf{i}\\ 0.5\mathbf{j}& 0.5\mathbf{i}+\mathbf{j}-\mathbf{k}\end{array}\right),\hfill \\ \hfill & F_2=\left(\begin{array}{cc}\mathbf{k}& \mathbf{j}\\ \mathbf{i}& \mathbf{i}\end{array}\right),H_1=\left(\begin{array}{cc}1-\mathbf{i}& 1+\mathbf{i}\\ 1-\mathbf{j}& 1-\mathbf{k}\end{array}\right),H_2=\left(\begin{array}{cc}\mathbf{i}& \mathbf{j}\\ \mathbf{j}& \mathbf{k}\end{array}\right),\hfill \\ & C_1=\left(\begin{array}{cc}-4.75+1.25\mathbf{i}-2.25\mathbf{j}+0.25\mathbf{k}& -0.5+\mathbf{i}-0.25\mathbf{j}+2.25\mathbf{k}\\ 0.75+\mathbf{i}-1.125\mathbf{j}+\mathbf{k}& 2\mathbf{i}+0.25\mathbf{j}\end{array}\right),\hfill \\ \hfill & C_2=\left(\begin{array}{cc}-6.25+2.75\mathbf{i}-0.75\mathbf{j}+2\mathbf{k}& 1+2.5\mathbf{i}-3\mathbf{j}+2.25\mathbf{k}\\ -1-\mathbf{j}-0.75\mathbf{k}& \mathbf{i}-0.5\mathbf{j}+0.75\mathbf{k}\end{array}\right),\hfill \\ & C_3=\left(\begin{array}{cc}0.5+3.5\mathbf{i}+\mathbf{j}-0.5\mathbf{k}& -1.5+2.75\mathbf{i}+2.5\mathbf{j}+\mathbf{k}\\ -1+0.5\mathbf{i}+1.25\mathbf{j}+1.25\mathbf{k}& 2+0.75\mathbf{j}+1.75\mathbf{k}\end{array}\right).\hfill \end{array}$$

be given.

Based on MATLAB 2023 (R2023a) and Algorithm 1, we have that

$$r\left(\begin{array}{c}{U}_1\\ U_2\end{array}\right)=48=12nt,{\theta }_2=0.7532\times {10}^{-13}.$$

We use the following:

$${\tilde{X}}_1=\left(\begin{array}{cc}1+0.5\mathbf{j}+2\mathbf{k}& 1+\mathbf{i}-\mathbf{j}\\ 0.5+\mathbf{i}-\mathbf{j}& 0.5\mathbf{k}\end{array}\right),{\tilde{X}}_2=\left(\begin{array}{cc}0.5+\mathbf{i}+2\mathbf{j}& \mathbf{i}+\mathbf{k}\\ 0& 0.25\mathbf{j}\end{array}\right),{\tilde{X}}_3=\left(\begin{array}{cc}1+\mathbf{j}& 0.25\mathbf{k}\\ 0.5\mathbf{i}& 0\end{array}\right).$$

Let

$$\begin{array}{cc}\hfill & {\psi }_{C_1}={\psi }_{A_1}S\left({\tilde{X}}_1\right)S\left(B_1\right)+{\psi }_{A_1}S\left({\tilde{X}}_2\right)S\left(B_2\right)+{\psi }_{A_2}S\left({\tilde{X}}_3\right)S\left(B_2\right),\hfill \\ \hfill & {\psi }_{C_2}={\psi }_{E_1}S\left({\tilde{X}}_1\right)S\left(F_1\right)+{\psi }_{E_1}S\left({\tilde{X}}_2\right)S\left(F_2\right)+{\psi }_{E_2}S\left({\tilde{X}}_3\right)S\left(F_2\right),\hfill \\ \hfill & {\psi }_{C_3}={\psi }_{G_1}S\left({\tilde{X}}_1\right)S\left(H_1\right)+{\psi }_{G_1}S\left({\tilde{X}}_2\right)S\left(H_2\right)+{\psi }_{G_2}S\left({\tilde{X}}_3\right)S\left(H_2\right).\hfill \end{array}$$

In addition, we can also obtain that ${\theta }_1=1.0436\times {10}^{-13},$ and ${\theta }_3=0.6579\times {10}^{-13}.$ According to Algorithm 1, System (1) has a unique solution that satisfies the following:

$$\left\{\begin{array}{c}\left(\begin{array}{ccc}{X}_1& X_2& X_3\end{array}\right)|\left(\begin{array}{c}vec\left(\widehat{X_1}\right)\\ vec\left(\widehat{X_2}\right)\\ vec\left(\widehat{X_3}\right)\end{array}\right)={\left(\begin{array}{c}{U}_1\\ U_2\end{array}\right)}^{†}\xi \end{array}\right\},$$

where

$${\left(\begin{array}{c}{U}_1\\ U_2\end{array}\right)}^{†}\xi =\left(\begin{array}{c}{V}_1\\ V_2\\ V_3\\ V_4\end{array}\right),$$

$$V_1=\left(\begin{array}{c}-8.9336\\ 5.2118\\ 16.8040\\ 2.5231\\ 2.2664\\ 8.0933\\ -1.5101\\ -5.5696\\ -0.1943\\ -2.4662\\ 8.8664\\ 3.3981\end{array}\right),V_2=\left(\begin{array}{c}3.6108\\ 8.3739\\ 3.2151\\ -1.9446\\ 0.2637\\ 5.3229\\ 5.8168\\ -3.9265\\ 9.3746\\ 2.2365\\ -3.1720\\ -6.1887\end{array}\right),V_3=\left(\begin{array}{c}-5.8453\\ 0.9736\\ -3.9148\\ -8.4436\\ -2.2233\\ 3.6062\\ -1.7516\\ -3.0795\\ -2.1770\\ 1.8020\\ -3.5136\\ 4.6386\end{array}\right),V_4=\left(\begin{array}{c}4.7273\\ -4.6023\\ -10.0539\\ 5.6789\\ -5.0584\\ 2.9334\\ -1.4736\\ 1.8486\\ 7.4243\\ -5.5493\\ -3.4163\\ 2.2913\end{array}\right).$$

Algorithm 1 For the system (1)

1: Input the values for matrices $A_i,B_i,E_i,F_i,G_i,H_i(i=1,2)$ and $C_1,C_2,C_3.$ 2: Calculate $U_1,U_2,\widehat{G},\widehat{J},\widehat{R},{\widehat{Q}}_1,{\widehat{Q}}_2,{\widehat{Q}}_{3}and\xi$. 3: If both (9) and (12) hold, then calculate $\left(\begin{array}{ccc}{X}_{1\eta }& X_{2\eta }& X_{3\eta }\end{array}\right)\in \Omega accordingto\left(14\right).$ 4: If (12) holds, then calculate $\left(\begin{array}{ccc}{X}_{1\eta }& X_{2\eta }& X_{3\eta }\end{array}\right)\in \Omega by\left(13\right).$ 5: Output $\left(\begin{array}{ccc}{X}_{1\eta }& X_{2\eta }& X_{3\eta }\end{array}\right).$

Example 2. 

Let

$$\begin{array}{cc}\hfill & A_2=\left(\begin{array}{cc}0.5\mathbf{i}& 0.5\mathbf{j}\\ 0& \mathbf{i}\end{array}\right),B_2=\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right),B_1=\left(\begin{array}{cc}\mathbf{i}+\mathbf{j}& 0\\ 0& 0.5\mathbf{j}+\mathbf{k}\end{array}\right),\hfill \\ & G_1=\left(\begin{array}{cc}1& 1\\ 1& 1\end{array}\right),H_2=\left(\begin{array}{cc}0& 0\\ 0& 1\end{array}\right),F_1=\left(\begin{array}{cc}\mathbf{i}+0.5\mathbf{j}+\mathbf{k}& \mathbf{i}\\ 0.5\mathbf{i}-\mathbf{j}+\mathbf{k}& 0.5\mathbf{j}\end{array}\right),\hfill \\ \hfill & H_1=\left(\begin{array}{cc}\mathbf{i}+2\mathbf{j}+0.5\mathbf{k}& \mathbf{i}-\mathbf{j}+\mathbf{k}\\ \mathbf{i}+0.5\mathbf{j}& \mathbf{i}-0.5\mathbf{k}\end{array}\right),B_2=\left(\begin{array}{cc}\mathbf{i}& \mathbf{j}\\ \mathbf{k}& \mathbf{i}\end{array}\right),\hfill \\ & C_1=\left(\begin{array}{cc}-0.5+0.375\mathbf{i}& 0.5\mathbf{k}\\ -0.25\mathbf{j}& 0\end{array}\right),F_1=\left(\begin{array}{cc}1+\mathbf{i}& 1+\mathbf{i}\\ 0.5\mathbf{j}& 0.5\mathbf{i}+\mathbf{j}-\mathbf{k}\end{array}\right),\hfill \\ \hfill & C_2=\left(\begin{array}{cc}-1.5+0.375\mathbf{i}-4.75\mathbf{j}+1.5\mathbf{k}& 0.5+0.5\mathbf{i}-2.5\mathbf{j}\\ 1.5\mathbf{i}-0.75\mathbf{j}+0.5\mathbf{k}& 1.125+0.5\mathbf{i}-\mathbf{j}\end{array}\right),\hfill \\ & C_3=\left(\begin{array}{cc}1.625+7\mathbf{i}+2\mathbf{j}+1.5\mathbf{k}& -1.5+1.625\mathbf{i}+0.5\mathbf{j}+6.25\mathbf{k}\\ 1.625+7\mathbf{i}+2\mathbf{j}+1.5\mathbf{k}& -4+1.625\mathbf{i}+0.5\mathbf{j}+6.25\mathbf{k}\end{array}\right),\hfill \\ \hfill & A_1=0,B_2=-F_2=E_1,E_2=A_2,G_2=F_1.\hfill \end{array}$$

be given.

We use the following:

$${\tilde{X}}_1=\left(\begin{array}{cc}0.5+\mathbf{j}+\mathbf{k}& 1+\mathbf{i}+\mathbf{j}\\ 1-\mathbf{i}+\mathbf{k}& 0.25\mathbf{j}\end{array}\right),{\tilde{X}}_2=\left(\begin{array}{cc}1+3\mathbf{i}+\mathbf{j}& 2\mathbf{j}+\mathbf{k}\\ 0& 0.5\mathbf{i}\end{array}\right),{\tilde{X}}_3=\left(\begin{array}{cc}0.5+\mathbf{i}& \mathbf{j}\\ 0.25\mathbf{k}& 0\end{array}\right).$$

Let

$$\begin{array}{cc}\hfill & {\psi }_{C_1}={\psi }_{A_1}S\left({\tilde{X}}_1\right)S\left(B_1\right)+{\psi }_{A_1}S\left({\tilde{X}}_2\right)S\left(B_2\right)+{\psi }_{A_2}S\left({\tilde{X}}_3\right)S\left(B_2\right),\hfill \\ \hfill & {\psi }_{C_2}={\psi }_{E_1}S\left({\tilde{X}}_1\right)S\left(F_1\right)+{\psi }_{E_1}S\left({\tilde{X}}_2\right)S\left(F_2\right)+{\psi }_{E_2}S\left({\tilde{X}}_3\right)S\left(F_2\right),\hfill \\ \hfill & {\psi }_{C_3}={\psi }_{G_1}S\left({\tilde{X}}_1\right)S\left(H_1\right)+{\psi }_{G_1}S\left({\tilde{X}}_2\right)S\left(H_2\right)+{\psi }_{G_2}S\left({\tilde{X}}_3\right)S\left(H_2\right).\hfill \end{array}$$

Based on MATLAB 2023 (R2023a) and Algorithm 1, we have that

$$r\left(\begin{array}{c}{U}_1\\ U_2\end{array}\right)=39<12n\times t,{\theta }_2=8.6892\times {10}^{-14}.$$

Therefore, we can easily see that System (1) is consistent. In addition, we can also obtain that ${\theta }_1=2.2730\times {10}^{-14},$ and ${\theta }_3=3.8286\times {10}^{-14}.$ Consequently, System (1) has infinite solutions $[X_1,X_2,X_3]\in \Omega$ and an approximate solution $\left[X_{1\eta },X_{2\eta },X_{3\eta }\right]\in \Omega .$

7. Conclusions

In this work, we established the necessary and sufficient conditions for the solvability of System (1) over the commutative quaternion ring, including the existence of a Hermitian solution. Moreover, we derived formulas for both the general and Hermitian solutions. As a practical application of our findings, we demonstrated a reducible solution for the classical system (2) over the commutative quaternion ring. In future work, we may extend this framework by exploring similar systems within the context of split quaternions.