Three Symmetrical Systems of Coupled Sylvester-like Quaternion Matrix Equations

Abstract

The current study investigates the solvability conditions and the general solution of three symmetrical systems of coupled Sylvester-like quaternion matrix equations. Accordingly, the necessary and sufficient conditions for the consistency of these systems are determined, and the general solutions of the systems are thereby deduced. An algorithm and a numerical example are constructed over the quaternions to validate the results of this paper.

1. Introduction

Sylvester-like matrix equations and their generalizations have applications in many scientific fields such as graph theory [1], image processing [2], output feedback control [3], neural networks [4], eigenvalue assignment problems [5], robust control [6] and so on (for example, see [7,8,9,10,11,12,13,14,15,16,17]). The iterative methods of solving linear and nonlinear Sylvester-like matrix equations are built on numerical solution methods [18,19,20,21,22,23,24,25,26,27]. So far, there has not been fruitful information on the solvability conditions and the general solution to systems of mixed and coupled Sylvester-like quaternion matrix equations. For the sake of understanding, analyzing, and developing theoretical studies of generalized systems of Sylvester coupled matrix equations and their applications, we, in this paper, consider the solvability conditions and the general solutions of the following three symmetrical systems of coupled Sylvester-like quaternion matrix equations:

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}\hfill & A_{1}V=C_1,V{B}_1=C_2,\hfill \\ \hfill & A_{3}X+Y{B}_3=C_3,\hfill \\ \hfill & A_{2}Y+Z{B}_2+A_{5}V{B}_5=C_5,\hfill \\ \hfill & A_{4}W+Z{B}_4=C_4,\hfill \end{array}\right.\end{array}$$

(1)

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}\hfill & A_{1}V=C_1,V{B}_1=C_2,\hfill \\ \hfill & A_{3}X+Y{B}_3=C_3,\hfill \\ \hfill & A_{2}Z+Y{B}_2+A_{5}V{B}_5=C_5,\hfill \\ \hfill & A_{4}Z+W{B}_4=C_4,\hfill \end{array}\right.\end{array}$$

(2)

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}\hfill & A_{1}V=C_1,V{B}_1=C_2,\hfill \\ \hfill & A_{3}X+Y{B}_3=C_3,\hfill \\ \hfill & A_{2}Y+Z{B}_2+A_{5}V{B}_5=C_5,\hfill \\ \hfill & A_{4}Z+W{B}_4=C_4,\hfill \end{array}\right.\end{array}$$

(3)

where $A_i,B_i$ and $C_i$$(i=\overline{1,5})$ are the given matrices with compatible dimensions, while $X,Y,Z,W,$ and V are the unknown matrices over $\mathbb{H}$. This work is motivated by a variety of well-known results on the solvability conditions and the general solutions of linear matrix equations in the literature, especially the findings of the following generalized Sylvester matrix equation:

$$\begin{array}{cc}\hfill & A_{1}X+Y{B}_1=C_1.\hfill \end{array}$$

(4)

Baksalary and Kala [28] came up with the general solution to (4). Lee and Vu [29] conducted a thorough examination of the general solution and solvability conditions for the following coupled system of matrix equations:

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

(5)

where $X,Y$ and Z are unknowns over the complex field with compatible sizes. Wang and He [30] derived the solvability conditions and the general solution for the following systems of symmetrical coupled Sylvester-like matrix equations:

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}\hfill & A_{1}X+Y{B}_1=C_1,\hfill \\ \hfill & A_{2}Y+Z{B}_2=C_2,\hfill \\ \hfill & A_{3}W+Z{B}_3=C_3,\hfill \end{array}\right.\left\{\begin{array}{cc}\hfill & A_{1}X+Y{B}_1=C_1,\hfill \\ \hfill & A_{2}Z+Y{B}_2=C_2,\hfill \\ \hfill & A_{3}Z+W{B}_3=C_3,\hfill \end{array}\right.\left\{\begin{array}{cc}\hfill & A_{1}X+Y{B}_1=C_1,\hfill \\ \hfill & A_{2}Y+Z{B}_2=C_2,\hfill \\ \hfill & A_{3}Z+W{B}_3=C_3.\hfill \end{array}\right.\end{array}$$

(6)

Meanwhile, Liu et al. [31] derive the determinantal representations of the general solution to (6). The solvability conditions and the general solution formulas for the matrix equations:

$$\begin{array}{cc}\hfill & AX=B,XC=D,\hfill \end{array}$$

(7)

and

$$\begin{array}{cc}\hfill & A_1{X}_1+Y_1{B}_1+C_{1}Z{D}_1=E_1\hfill \end{array}$$

(8)

are given by [32,33], respectively, which have a fundamental role in inferring the main results of this paper. Here, we provide proper generalizations to the systems of Sylvester-like matrix Equations (4)–(8). In particular, we reproduce the results of [11,29,33,34], see Section 3.1. In [30,33], He and Wang presented the solvability conditions and the general solution for the following Sylvester-like matrix equation:

$$\begin{array}{cc}\hfill & A_1{X}_1+X_2{B}_1+C_3{X}_3{D}_3+C_4{X}_4{D}_4=E_1.\hfill \end{array}$$

(9)

They presented these findings in terms of the Moore–Penrose inverse of certain given matrices. The resulting equation has wide applications in the linear matrix equations field, see [7,34,35,36,37,38,39,40]. He and Meng [41] derived the rank equalities solvability conditions for the following quaternion matrix equation with two different restrictions:

$$\begin{array}{cc}\hfill & A_1{Z}_1{B}_1+A_2{Z}_2{B}_2+A_3{Z}_3{B}_3=C_1,D{Z}_1=F,Z_{1}H=G.\hfill \end{array}$$

(10)

The solvability conditions and the general solution to the Sylvester-like matrix equation

$$\begin{array}{cc}\hfill & {\stackrel{`}{A}}_1{X}_1+Y_1\stackrel{`}{B_1}+{\stackrel{`}{A}}_2{Z}_1{\stackrel{`}{B}}_2+{\stackrel{`}{A}}_3{Z}_2{\stackrel{`}{B}}_3+{\stackrel{`}{A}}_4{Z}_3{\stackrel{`}{B}}_4={\stackrel{`}{C}}_1\hfill \end{array}$$

(11)

were given by [42], which is crucial to the proof in this study.

We organize this paper as follows. Some definitions and lemmas are reviewed in Section 2. In Section 3, we carry out the solvability conditions and the general solution of (1). In Section 4, we consider similarly the systems (2) and (3). To close this paper, we give a conclusion in Section 5.

2. Preliminaries

We denote the real number field by $\mathbb{R},$ while $\mathbb{H}$ stands for the quaternion algebra, which considered as a division ring [12,43,44]. ${\mathbb{H}}^{l\times k}$ expresses the set of all $l\times k$ matrices over $\mathbb{H}$. The symbol $A^{*}$ represents the conjugate transpose of A. The Moore–Penrose inverse of $A\in {\mathbb{H}}^{l\times k}$ is denoted by $A^{†}\in {\mathbb{H}}^{k\times l}$. Furthermore, $L_A=I-A^{†}A$, $R_A=I-A{A}^{†}$ stand for the two projectors of A and $r\left(A\right)$ represents the rank of the matrix A.

Lemma 1

([45]).    Let $A\in {\mathbb{H}}^{m\times n},$$B\in {\mathbb{H}}^{m\times k}$ and $C\in {\mathbb{H}}^{l\times n}$ be given. Then

$$\begin{array}{cc}\hfill & \left(1\right)r\left(A\right)+r\left(R_{A}B\right)=r\left(B\right)+r\left(R_{B}A\right)=r\left(\begin{array}{cc}A& B\end{array}\right),\hfill \\ \hfill & \left(2\right)r\left(A\right)+r\left(C{L}_A\right)=r\left(C\right)+r\left(A{L}_C\right)=r\left(\begin{array}{c}A\\ C\end{array}\right).\hfill \end{array}$$

Lemma 2

([32]). Let A, B, C and D be given. Then the quaternion matrix Equation (7) is solvable if and only if

$$\begin{array}{cc}\hfill & R_{A}B=0,D{L}_C=0,AD=BC.\hfill \end{array}$$

In this case, the general solution of (7) can be expressed as

$$\begin{array}{cc}\hfill & X=A^{†}B+L_{A}D{C}^{†}+L_{A}V{R}_C,\hfill \end{array}$$

where V is an arbitrary matrix over $\mathbb{H}$ with suitable size.

Lemma 3

([28]). Let $A_1,$ $B_1$ and $C_1$ be given. Then the quaternion matrix Equation (4) is solvable if and only if

$$\begin{array}{cc}\hfill & R_{A_1}{C}_1{L}_{B_1}=0.\hfill \end{array}$$

In this case, the general solution of (4) can be expressed as

$$\begin{array}{cc}\hfill & X=A_1^{†}{C}_1+U_1{B}_1+L_{A_1}{U}_2,\hfill \\ \hfill & Y=R_{A_1}{C}_1{B}_1^{†}+A_1{U}_1+U_3{R}_{B_1},\hfill \end{array}$$

where $U_i,(i=\overline{1,3})$ are arbitrary matrices over $\mathbb{H}$ with appropriate sizes.

Lemma 4

([42]). Let $\stackrel{`}{A_i},$$\stackrel{`}{B_i}$$(i=\overline{1,4})$ and $\stackrel{`}{C_1}$ be given. For simplicity, set

$$\begin{array}{cc}\hfill & A_{ii}=R_{\stackrel{`}{A_1}}\stackrel{`}{A_{i+1}},B_{ii}=\stackrel{`}{B_{i+1}}{L}_{\stackrel{`}{B_1}}(i=\overline{1,3}),M_1=R_{A_{11}}{A}_{22},S_1=A_{22}{L}_{M_1},\hfill \\ \hfill & N_1=B_{22}{L}_{B_{11}},T_1=R_{\stackrel{`}{A_1}}\stackrel{`}{C_1}{L}_{\stackrel{`}{B_1}},T=T_1-A_{33}{Z}_3{B}_{33},P=\left(\begin{array}{cc}{A}_{11},& A_{22}\end{array}\right),\hfill \\ \hfill & Q=\left(\begin{array}{c}{B}_{11}\\ B_{22}\end{array}\right),H_1=B_{33},H_2=B_{33}{L}_{B_{22}},H_3=B_{33}{L}_{B_{11}},H_4=B_{33}{L}_Q,G_1=R_P{A}_{33},\hfill \\ \hfill & G_2=R_{A_{11}}{A}_{33},G_3=R_{A_{22}}{A}_{33},G_4=A_{33},L_1=R_P{T}_1,L_2=R_{A_{11}}{T}_1{L}_{B_{22}},\hfill \\ \hfill & L_3=R_{A_{22}}{T}_1{L}_{B_{11}},L_4=T_1{L}_Q,\widehat{A_1}=\left(\begin{array}{cc}{L}_{G_2}& L_{G_4}\end{array}\right),\widehat{B_1}=\left(\begin{array}{c}{R}_{H_1}\\ R_{H_3}\end{array}\right),\widehat{C_1}=L_{G_1},\hfill \\ \hfill & \widehat{D_1}=R_{H_2},\widehat{A_2}=R_{\widehat{A_1}}\widehat{C_1},\widehat{B_2}=\widehat{D_1}{L}_{\widehat{B_1}},\widehat{C_2}=L_{G_3},\widehat{D_2}=R_{H_4},\widehat{A_3}=R_{\widehat{A_1}}\widehat{C_2},\hfill \\ \hfill & \widehat{B_3}=\widehat{D_2}{L}_{\widehat{B_1}},M=R_{\widehat{A_2}}\widehat{A_3},N=\widehat{B_3}{L}_{\widehat{B_2}},S=\widehat{A_3}{L}_M,Q_{11}^0=G_1^{†}{L}_1{H}_1^{†}+L_{G_1}{G}_2^{†}{L}_2{H}_2^{†},\hfill \\ & Q_{22}^0=G_3^{†}{L}_3{H}_3^{†}+L_{G_3}{G}_4^{†}{L}_4{H}_4^{†},\overline{Q}=Q_{22}^0-Q_{11}^0,E=R_{\widehat{A_1}}\overline{Q}{L}_{\widehat{B_1}}.\hfill \end{array}$$

(12)

Then the following statements are equivalent:

(1) The matrix Equation (11) is solvable.

(2)

$$\begin{array}{cc}\hfill & r\left(\begin{array}{ccccc}\stackrel{`}{C_1}& \stackrel{`}{A_2}& \stackrel{`}{A_3}& \stackrel{`}{A_4}& \stackrel{`}{A_1}\\ \stackrel{`}{B_1}& 0& 0& 0& 0\end{array}\right)=r\left(\stackrel{`}{B_1}\right)+r\left(\begin{array}{cccc}\stackrel{`}{A_2}& \stackrel{`}{A_3}& \stackrel{`}{A_4}& \stackrel{`}{A_1}\end{array}\right),\hfill \end{array}$$

(13)

$$\begin{array}{cc}\hfill & r\left(\begin{array}{cc}\stackrel{`}{C_1}& \stackrel{`}{A_1}\\ \stackrel{`}{B_2}& 0\\ \stackrel{`}{B_3}& 0\\ \stackrel{`}{B_4}& 0\\ \stackrel{`}{B_1}& 0\end{array}\right)=r\left(\stackrel{`}{A_1}\right)+r\left(\begin{array}{c}\stackrel{`}{B_2}\\ \stackrel{`}{B_3}\\ \stackrel{`}{B_4}\\ \stackrel{`}{B_1}\end{array}\right),\hfill \end{array}$$

(14)

$$\begin{array}{cc}\hfill & r\left(\begin{array}{cccc}\stackrel{`}{C_1}& \stackrel{`}{A_2}& \stackrel{`}{A_3}& \stackrel{`}{A_1}\\ \stackrel{`}{B_4}& 0& 0& 0\\ \stackrel{`}{B_1}& 0& 0& 0\end{array}\right)=r\left(\begin{array}{ccc}\stackrel{`}{A_2}& \stackrel{`}{A_3}& \stackrel{`}{A_1}\end{array}\right)+r\left(\begin{array}{c}\stackrel{`}{B_4}\\ \stackrel{`}{B_1}\end{array}\right),\hfill \end{array}$$

(15)

$$\begin{array}{cc}\hfill & r\left(\begin{array}{cccc}\stackrel{`}{C_1}& \stackrel{`}{A_2}& \stackrel{`}{A_4}& \stackrel{`}{A_1}\\ \stackrel{`}{B_3}& 0& 0& 0\\ \stackrel{`}{B_1}& 0& 0& 0\end{array}\right)=r\left(\begin{array}{ccc}\stackrel{`}{A_2}& \stackrel{`}{A_4}& \stackrel{`}{A_1}\end{array}\right)+r\left(\begin{array}{c}\stackrel{`}{B_3}\\ \stackrel{`}{B_1}\end{array}\right),\hfill \end{array}$$

(16)

$$\begin{array}{cc}\hfill & r\left(\begin{array}{cccc}\stackrel{`}{C_1}& \stackrel{`}{A_3}& \stackrel{`}{A_4}& \stackrel{`}{A_1}\\ \stackrel{`}{B_2}& 0& 0& 0\\ \stackrel{`}{B_1}& 0& 0& 0\end{array}\right)=r\left(\begin{array}{ccc}\stackrel{`}{A_3}& \stackrel{`}{A_4}& \stackrel{`}{A_1}\end{array}\right)+r\left(\begin{array}{c}\stackrel{`}{B_2}\\ \stackrel{`}{B_1}\end{array}\right),\hfill \end{array}$$

(17)

$$\begin{array}{cc}\hfill & r\left(\begin{array}{ccc}\stackrel{`}{C_1}& \stackrel{`}{A_4}& \stackrel{`}{A_1}\\ \stackrel{`}{B_2}& 0& 0\\ \stackrel{`}{B_3}& 0& 0\\ \stackrel{`}{B_1}& 0& 0\end{array}\right)=r\left(\begin{array}{cc}\stackrel{`}{A_4}& \stackrel{`}{A_1}\end{array}\right)+r\left(\begin{array}{c}\stackrel{`}{B_2}\\ \stackrel{`}{B_3}\\ \stackrel{`}{B_1}\end{array}\right),\hfill \end{array}$$

(18)

$$\begin{array}{cc}\hfill & r\left(\begin{array}{ccc}\stackrel{`}{C_1}& \stackrel{`}{A_3}& \stackrel{`}{A_1}\\ \stackrel{`}{B_2}& 0& 0\\ \stackrel{`}{B_4}& 0& 0\\ \stackrel{`}{B_1}& 0& 0\end{array}\right)=r\left(\begin{array}{cc}\stackrel{`}{A_3}& \stackrel{`}{A_1}\end{array}\right)+r\left(\begin{array}{c}\stackrel{`}{B_2}\\ \stackrel{`}{B_4}\\ \stackrel{`}{B_1}\end{array}\right),\hfill \end{array}$$

(19)

$$\begin{array}{cc}\hfill & r\left(\begin{array}{ccc}\stackrel{`}{C_1}& \stackrel{`}{A_2}& \stackrel{`}{A_1}\\ \stackrel{`}{B_3}& 0& 0\\ \stackrel{`}{B_4}& 0& 0\\ \stackrel{`}{B_1}& 0& 0\end{array}\right)=r\left(\begin{array}{cc}\stackrel{`}{A_2}& \stackrel{`}{A_1}\end{array}\right)+r\left(\begin{array}{c}\stackrel{`}{B_3}\\ \stackrel{`}{B_4}\\ \stackrel{`}{B_1}\end{array}\right),\hfill \end{array}$$

(20)

$$\begin{array}{cc}\hfill & r\left(\begin{array}{ccccccc}\stackrel{`}{C_1}& \stackrel{`}{A_2}& \stackrel{`}{A_1}& 0& 0& 0& \stackrel{`}{A_4}\\ \stackrel{`}{B_3}& 0& 0& 0& 0& 0& 0\\ \stackrel{`}{B_1}& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& -\stackrel{`}{C_1}& \stackrel{`}{A_3}& \stackrel{`}{A_1}& \stackrel{`}{A_4}\\ 0& 0& 0& \stackrel{`}{B_2}& 0& 0& 0\\ 0& 0& 0& \stackrel{`}{B_1}& 0& 0& 0\\ \stackrel{`}{B_4}& 0& 0& \stackrel{`}{B_4}& 0& 0& 0\end{array}\right)=r\left(\begin{array}{cc}\stackrel{`}{B_3}& 0\\ \stackrel{`}{B_1}& 0\\ 0& \stackrel{`}{B_2}\\ 0& \stackrel{`}{B_1}\\ \stackrel{`}{B_4}& \stackrel{`}{B_4}\end{array}\right)+r\left(\begin{array}{ccccc}\stackrel{`}{A_2}& \stackrel{`}{A_1}& 0& 0& \stackrel{`}{A_4}\\ 0& 0& \stackrel{`}{A_3}& \stackrel{`}{A_1}& \stackrel{`}{A_4}\end{array}\right).\hfill \end{array}$$

(21)

(3)

$$\begin{array}{cc}\hfill & R_{G_i}{L}_i=0,L_i{L}_{H_i}=0(i=\overline{1,4}),R_{\widehat{A_3}}E{L}_{\widehat{B_2}}=0.\hfill \end{array}$$

In this case, the general solution of (11) can be expressed as

$$\begin{array}{cc}\hfill & X_1={\stackrel{`}{A_1}}^{†}(\stackrel{`}{C_1}-\stackrel{`}{A_2}{Z}_1\stackrel{`}{B_2}-\stackrel{`}{A_3}{Z}_2\stackrel{`}{B_3}-\stackrel{`}{A_4}{Z}_3\stackrel{`}{B_4})-P_1\stackrel{`}{B_1}+L_{\stackrel{`}{A_1}}{P}_2,\hfill \\ \hfill & Y_1=R_{\stackrel{`}{A_1}}(\stackrel{`}{C_1}-\stackrel{`}{A_2}{Z}_1\stackrel{`}{B_2}-\stackrel{`}{A_3}{Z}_2\stackrel{`}{B_3}-\stackrel{`}{A_4}{Z}_3\stackrel{`}{B_4}){\stackrel{`}{B_1}}^{†}+\stackrel{`}{A_1}{P}_1+P_3{R}_{\stackrel{`}{B_1}},\hfill \\ \hfill & Z_1=A_{11}^{†}T{B}_{11}^{†}-A_{11}^{†}{A}_{22}{M}_1^{†}T{B}_{11}^{†}-A_{11}^{†}{S}_1{A}_{22}^{†}T{N}_1^{†}{B}_{22}{B}_{11}^{†}\hfill \\ \hfill & -A_{11}^{†}{S}_1{P}_4{R}_{N_1}{B}_{22}{B}_{11}^{†}+L_{A_{11}}{P}_5+P_6{R}_{B_{11}},\hfill \\ \hfill & Z_2=M_1^{†}T{B}_{22}^{†}+S_1^{†}{S}_1{A}_{22}^{†}T{N}_1^{†}+L_{M_1}{L}_{S_1}{P}_7+P_8{R}_{B_{22}}+L_{M_1}{U}_4{R}_{N_1},\hfill \end{array}$$

$$\begin{array}{cc}\hfill & Z_3=Q_{11}^0+L_{G_2}{J}_1+J_2{R}_{H_1}+L_{G_1}{J}_3{R}_{H_2},\hfill \\ \hfill & or{Z}_3=Q_{22}^0-L_{G_4}{K}_1-K_2{R}_{H_3}-L_{G_3}{K}_3{R}_{H_4},\hfill \end{array}$$

where $P_i$$(i=\overline{1,8})$ are arbitrary matrices over $\mathbb{H}$ with fit sizes and

$$\begin{array}{cc}\hfill & J_1=\left(\begin{array}{cc}{I}_m& 0\end{array}\right)[{\widehat{A_1}}^{†}(\overline{Q}-\widehat{C_1}{J}_3\widehat{D_1}-\widehat{C_2}{K}_3\widehat{D_2})-U_{11}\widehat{B_1}+L_{\widehat{A_1}}{U}_{12}],\hfill \\ \hfill & K_1=\left(\begin{array}{cc}0& I_m\end{array}\right)[{\widehat{A_1}}^{†}(\overline{Q}-\widehat{C_1}{J}_3\widehat{D_1}-\widehat{C_2}{K}_3\widehat{D_2})-U_{11}\widehat{B_1}+L_{\widehat{A_1}}{U}_{12}],\hfill \\ \hfill & J_2=[R_{\widehat{A_1}}(\overline{Q}-\widehat{C_1}{J}_3\widehat{D_1}-\widehat{C_2}{K}_3\widehat{D_2}){\widehat{B_1}}^{†}+\widehat{A_1}{U}_{11}+U_{21}{R}_{\widehat{B_1}}]\left(\begin{array}{c}{I}_n\\ 0\end{array}\right),\hfill \\ \hfill & K_2=[R_{\widehat{A_1}}(\overline{Q}-\widehat{C_1}{J}_3\widehat{D_1}-\widehat{C_2}{K}_3\widehat{D_2}){\widehat{B_1}}^{†}+\widehat{A_1}{U}_{11}+U_{21}{R}_{\widehat{B_1}}]\left(\begin{array}{c}0\\ I_n\end{array}\right),\hfill \\ \hfill & J_3={\widehat{A_2}}^{†}\overline{Q}{\widehat{B_2}}^{†}-{\widehat{A_2}}^{†}\widehat{A_3}{M}^{†}\overline{Q}{\widehat{B_2}}^{†}-{\widehat{A_2}}^{†}S{\widehat{A_3}}^{†}\overline{Q}{N}^{†}\widehat{B_3}{\widehat{B_2}}^{†}\hfill \\ \hfill & -{\widehat{A_2}}^{†}S{U}_{31}{R}_N\widehat{B_3}{\widehat{B_2}}^{†}+L_{\widehat{A_2}}{U}_{32}+U_{33}{R}_{\widehat{B_2}},\hfill \\ \hfill & K_3=M^{†}\overline{Q}{\widehat{B_3}}^{†}+S^{†}S{\widehat{A_3}}^{†}\overline{Q}{N}^{†}+L_M{L}_S{U}_{41}+U_{42}{R}_{\widehat{B_3}}+L_M{U}_{31}{R}_N,\hfill \end{array}$$

where $U_{11}$, $U_{12}$, $U_{21}$, $U_{31}$, $U_{32}$, $U_{33}$, $U_{41}$ and $U_{42}$ are arbitrary quaternion matrices, m and n are the number of columns of ${\stackrel{`}{A}}_4$ and the number of rows of ${\stackrel{`}{B}}_4$, respectively.

3. The Solvability Conditions and an Expression of the General Solution to (1)

Theorem 1.

Let $A_i,$$B_i$ and $C_i$$(i=\overline{1,5})$ be given. For simplicity, consider (12) such that

$$\begin{array}{cc}\hfill & \stackrel{`}{A_1}=A_2{A}_3,\stackrel{`}{B_1}=R_{B_4}{B}_2,\stackrel{`}{A_2}=A_2,\stackrel{`}{B_2}=R_{B_3},\stackrel{`}{A_3}=A_4,\stackrel{`}{B_3}=B_2,\stackrel{`}{A_4}=A_5{L}_{A_1},\hfill \\ \hfill & \stackrel{`}{B_4}=R_{B_1}{B}_5,\stackrel{`}{C_1}=C_5-[A_2{R}_{A_3}{C}_3{B}_3^{†}+R_{A_4}{C}_4{B}_4^{†}{B}_2+A_5(A_1^{†}{C}_1+L_{A_1}{C}_2{B}_1^{†})B_5].\hfill \end{array}$$

(22)

Then the following statements are equivalent:

(1) The system (1) is solvable.

(2)

$$\begin{array}{cc}\hfill & R_{A_1}{C}_1=0,C_2{L}_{B_1}=0,A_1{C}_2=C_1{B}_1,R_{A_i}{C}_i{L}_{B_i}=0(i=3,4),\hfill \\ \hfill & R_{G_i}{L}_i=0,L_i{L}_{H_i}=0(i=\overline{1,4}),R_{\widehat{A_3}}E{L}_{\widehat{B_2}}=0.\hfill \end{array}$$

(23)

(3)

$$\begin{array}{cc}\hfill & r\left(\begin{array}{cc}{C}_1& A_1\end{array}\right)=r\left(A_1\right),r\left(\begin{array}{c}{C}_2\\ B_1\end{array}\right)=r\left(B_1\right),A_1{C}_2=C_1{B}_1,\hfill \end{array}$$

(24)

$$\begin{array}{cc}\hfill & r\left(\begin{array}{cc}{C}_i& A_i\\ B_i& 0\end{array}\right)=r\left(\begin{array}{c}{A}_i\end{array}\right)+r\left(\begin{array}{c}{B}_i\end{array}\right),(i=3,4),\hfill \end{array}$$

(25)

$$\begin{array}{cc}\hfill & r\left(\begin{array}{ccccc}{C}_5& C_4& A_2& A_4& A_5\\ C_1{B}_5& 0& 0& 0& A_1\\ B_2& B_4& 0& 0& 0\end{array}\right)=r\left(\begin{array}{cc}{B}_2& B_4\end{array}\right)+r\left(\begin{array}{ccc}{A}_5& A_2& A_4\\ A_1& 0& 0\end{array}\right),\hfill \end{array}$$

(26)

$$\begin{array}{cc}\hfill & r\left(\begin{array}{ccc}{C}_5{B}_3-A_2{C}_3& A_5{C}_2& A_2{A}_3\\ B_2{B}_3& 0& 0\\ B_5{B}_3& B_1& 0\end{array}\right)=r\left(\begin{array}{c}{A}_2{A}_3\end{array}\right)+r\left(\begin{array}{cc}{B}_2{B}_3& 0\\ B_5{B}_3& B_1\end{array}\right),\hfill \end{array}$$

(27)

$$\begin{array}{cc}\hfill & r\left(\begin{array}{ccccc}{C}_5& C_4& A_5{C}_2& A_2& A_4\\ B_5& 0& B_1& 0& 0\\ B_2& B_4& 0& 0& 0\end{array}\right)=r\left(\begin{array}{cc}{A}_2& A_4\end{array}\right)+r\left(\begin{array}{ccc}{B}_5& B_1& 0\\ B_2& 0& B_4\end{array}\right),\hfill \end{array}$$

(28)

$$\begin{array}{cc}\hfill & r\left(\begin{array}{ccc}{C}_5& A_2& A_5\\ B_2& 0& 0\\ C_1{B}_5& 0& A_1\end{array}\right)=r\left(\begin{array}{cc}{A}_2& A_5\\ 0& A_1\end{array}\right)+r\left(\begin{array}{c}{B}_2\end{array}\right),\hfill \end{array}$$

(29)

$$\begin{array}{cc}\hfill & r\left(\begin{array}{ccccc}{C}_5{B}_3-A_2{C}_3& A_5& A_4& A_2{A}_3& C_4\\ B_2{B}_3& 0& 0& 0& B_4\\ C_1{B}_5{B}_3& A_1& 0& 0& 0\end{array}\right)=r\left(\begin{array}{ccc}{A}_5& A_2{A}_3& A_4\\ A_1& 0& 0\end{array}\right)+r\left(\begin{array}{cc}{B}_2{B}_3& B_4\end{array}\right),\hfill \end{array}$$

(30)

$$\begin{array}{cc}\hfill & r\left(\begin{array}{ccc}{C}_5{B}_3-A_2{C}_3& A_5& A_2{A}_3\\ B_2{B}_3& 0& 0\\ C_1{B}_5{B}_3& A_1& 0\end{array}\right)=r\left(\begin{array}{cc}{A}_5& A_2{A}_3\\ A_1& 0\end{array}\right)+r\left(\begin{array}{c}{B}_2{B}_3\end{array}\right),\hfill \end{array}$$

(31)

$$\begin{array}{cc}\hfill & \begin{array}{c}r\left(\begin{array}{ccccc}{C}_5{B}_3-A_2{C}_3& A_5{C}_2& A_2{A}_3& A_4& C_4\\ B_5{B}_3& B_1& 0& 0& 0\\ B_2{B}_3& 0& 0& 0& B_4\end{array}\right)\hfill \\ =r\left(\begin{array}{ccc}{B}_5{B}_3& B_1& 0\\ B_2{B}_3& 0& B_4\end{array}\right)+r\left(\begin{array}{cc}{A}_4& A_2{A}_3\end{array}\right),\hfill \end{array}\hfill \end{array}$$

(32)

$$\begin{array}{cc}\hfill & r\left(\begin{array}{ccc}{C}_5& A_2& A_5{C}_2\\ B_2& 0& 0\\ B_5& 0& B_1\end{array}\right)=r\left(\begin{array}{c}{A}_2\end{array}\right)+r\left(\begin{array}{cc}{B}_2& 0\\ B_5& B_1\end{array}\right),\hfill \end{array}$$

(33)

$$\begin{array}{cc}\hfill & \begin{array}{c}r\left(\begin{array}{cccccccc}{C}_5& A_2& 0& 0& 0& A_5& 0& 0\\ B_2& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& -C_5{B}_3+A_2{C}_3& A_4& A_2{A}_3& A_5& -C_4& -A_5{C}_2\\ 0& 0& B_2{B}_3& 0& 0& 0& B_4& 0\\ B_5& 0& B_5{B}_3& 0& 0& 0& 0& B_1\\ C_1{B}_5& 0& 0& 0& 0& A_1& 0& 0\end{array}\right)\hfill \\ =r\left(\begin{array}{cccc}{B}_2& 0& 0& 0\\ 0& B_2{B}_3& B_4& 0\\ B_5& B_5{B}_3& 0& B_1\end{array}\right)+r\left(\begin{array}{cccc}{A}_2& 0& 0& A_5\\ 0& A_4& A_2{A}_3& A_5\\ 0& 0& 0& A_1\end{array}\right).\hfill \end{array}\hfill \end{array}$$

(34)

In this case, the general solution of (1) can be expressed as

$$\begin{array}{cc}\hfill & X=A_3^{†}{C}_3+U_1{B}_3+L_{A_3}{U}_2,\hfill \end{array}$$

(35)

$$\begin{array}{cc}\hfill & Y=R_{A_3}{C}_3{B}_3^{†}+A_3{U}_1+U_3{R}_{B_3},\hfill \end{array}$$

(36)

$$\begin{array}{cc}\hfill & W=A_4^{†}{C}_4+W_1{B}_4+L_{A_4}{W}_2,\hfill \end{array}$$

(37)

$$\begin{array}{cc}\hfill & Z=R_{A_4}{C}_4{B}_4^{†}+A_4{W}_1+W_3{R}_{B_4},\hfill \end{array}$$

(38)

$$\begin{array}{cc}\hfill & V=A_1^{†}{C}_1+L_{A_1}{C}_2{B}_1^{†}+L_{A_1}Q{R}_{B_1},\hfill \end{array}$$

(39)

where

$$\begin{array}{cc}\hfill & W_2={\stackrel{`}{A_1}}^{†}(\stackrel{`}{C_1}-\stackrel{`}{A_2}{W}_1\stackrel{`}{B_2}-\stackrel{`}{A_3}{U}_1\stackrel{`}{B_3}-\stackrel{`}{A_4}Q\stackrel{`}{B_4})-P_1\stackrel{`}{B_1}+L_{\stackrel{`}{A_1}}{P}_2,\hfill \\ \hfill & U_3=R_{\stackrel{`}{A_1}}(\stackrel{`}{C_1}-\stackrel{`}{A_2}{W}_1\stackrel{`}{B_2}-\stackrel{`}{A_3}{U}_1\stackrel{`}{B_3}-\stackrel{`}{A_4}Q\stackrel{`}{B_4}){\stackrel{`}{B_1}}^{†}+\stackrel{`}{A_1}{P}_1+P_3{R}_{\stackrel{`}{B_1}},\hfill \\ \hfill & W_1=A_{11}^{†}T{B}_{11}^{†}-A_{11}^{†}{A}_{22}{M}_1^{†}T{B}_{11}^{†}-A_{11}^{†}{S}_1{A}_{22}^{†}T{N}_1^{†}{B}_{22}{B}_{11}^{†}\hfill \\ \hfill & -A_{11}^{†}{S}_1{P}_4{R}_{N_1}{B}_{22}{B}_{11}^{†}+L_{A_{11}}{P}_5+P_6{R}_{B_{11}},\hfill \\ \hfill & U_1=M_1^{†}T{B}_{22}^{†}+S_1^{†}{S}_1{A}_{22}^{†}T{N}_1^{†}+L_{M_1}{L}_{S_1}{P}_7+P_8{R}_{B_{22}}+L_{M_1}{U}_4{R}_{N_1},\hfill \\ \hfill & Q=Q_{11}^0+L_{G_2}{J}_1+J_2{R}_{H_1}+L_{G_1}{J}_3{R}_{H_2},\hfill \\ \hfill & orQ=Q_{22}^0-L_{G_4}{K}_1-K_2{R}_{H_3}-L_{G_3}{K}_3{R}_{H_4},\hfill \\ \hfill & J_1=\left(\begin{array}{cc}{I}_m& 0\end{array}\right)[{\widehat{A_1}}^{†}(\overline{Q}-\widehat{C_1}{J}_3\widehat{D_1}-\widehat{C_2}{K}_3\widehat{D_2})-U_{11}\widehat{B_1}+L_{\widehat{A_1}}{U}_{12}],\hfill \\ \hfill & K_1=\left(\begin{array}{cc}0& I_m\end{array}\right)[{\widehat{A_1}}^{†}(\overline{Q}-\widehat{C_1}{J}_3\widehat{D_1}-\widehat{C_2}{K}_3\widehat{D_2})-U_{11}\widehat{B_1}+L_{\widehat{A_1}}{U}_{12}],\hfill \\ \hfill & J_2=[R_{\widehat{A_1}}(\overline{Q}-\widehat{C_1}{J}_3\widehat{D_1}-\widehat{C_2}{K}_3\widehat{D_2}){\widehat{B_1}}^{†}+\widehat{A_1}{U}_{11}+U_{21}{R}_{\widehat{B_1}}]\left(\begin{array}{c}{I}_n\\ 0\end{array}\right),\hfill \\ \hfill & K_2=[R_{\widehat{A_1}}(\overline{Q}-\widehat{C_1}{J}_3\widehat{D_1}-\widehat{C_2}{K}_3\widehat{D_2}){\widehat{B_1}}^{†}+\widehat{A_1}{U}_{11}+U_{21}{R}_{\widehat{B_1}}]\left(\begin{array}{c}0\\ I_n\end{array}\right),\hfill \end{array}$$

$$\begin{array}{cc}\hfill & \begin{array}{c}{J}_3={\widehat{A_2}}^{†}\overline{Q}{\widehat{B_2}}^{†}-{\widehat{A_2}}^{†}\widehat{A_3}{M}^{†}\overline{Q}{\widehat{B_2}}^{†}-{\widehat{A_2}}^{†}S{\widehat{A_3}}^{†}\overline{Q}{N}^{†}\widehat{B_3}{\widehat{B_2}}^{†}\hfill \\ -{\widehat{A_2}}^{†}S{U}_{31}{R}_N\widehat{B_3}{\widehat{B_2}}^{†}+L_{\widehat{A_2}}{U}_{32}+U_{33}{R}_{\widehat{B_2}},\hfill \end{array}\hfill \\ \hfill & K_3=M^{†}\overline{Q}{\widehat{B_3}}^{†}+S^{†}S{\widehat{A_3}}^{†}\overline{Q}{N}^{†}+L_M{L}_S{U}_{41}+U_{42}{R}_{\widehat{B_3}}+L_M{U}_{31}{R}_N,\hfill \end{array}$$

where $P_i$$(i=\overline{1,8})$, $U_2$, $W_2$, $U_{11}$, $U_{12}$, $U_{21}$, $U_{31}$, $U_{32}$, $U_{33}$, $U_{41}$ and $U_{42}$ are arbitrary matrices over $\mathbb{H}$, m and n are the number of columns of $A_1$ and the number of rows of $B_1$, respectively.

Proof of Theorem 1.

$\left(1\right)\iff \left(2\right):$ Under the conditions $R_{A_1}{C}_1=0$, $C_2{L}_{B_1}=0$, $A_1{C}_2=C_1{B}_1,$ the matrix equations $A_{1}V=C_1,V{B}_1=C_2$ have a common solution. By using Lemma 2, we have the general common solution as

$$\begin{array}{cc}\hfill & V=A_1^{†}{C}_1+L_{A_1}{C}_2{B}_1^{†}+L_{A_1}Q{R}_{B_1},\hfill \end{array}$$

(40)

where Q is an arbitrary matrix over $\mathbb{H}$ with suitable size. For the matrix equations

$$\begin{array}{cc}\hfill & A_{3}X+Y{B}_3=C_3,\hfill \end{array}$$

(41)

$$\begin{array}{cc}\hfill & A_{4}Z+W{B}_4=C_4,\hfill \end{array}$$

(42)

when the conditions $R_{A_i}{C}_i{L}_{B_i}=0(i=3,4)$ are met, we have that (41) and (42) are solvable, respectively. By using Lemma 3, their general solutions can be expressed as

$$\begin{array}{cc}\hfill & X=A_3^{†}{C}_3+U_1{B}_3+L_{A_3}{U}_2,Y=R_{A_3}{C}_3{B}_3^{†}+A_3{U}_1+U_3{R}_{B_3},\hfill \end{array}$$

(43)

$$\begin{array}{cc}\hfill & Z=A_4^{†}{C}_4+W_1{B}_4+L_{A_4}{W}_2,W=R_{A_4}{C}_4{B}_4^{†}+A_4{W}_1+W_3{R}_{B_4},\hfill \end{array}$$

(44)

where $W_i$ and $V_i$$(i=\overline{1,3})$ are arbitrary matrices with suitable sizes. Substituting (40), (43) and (44) into the equation

$$\begin{array}{cc}\hfill & A_{2}Z+Y{B}_2+A_{5}V{B}_5=C_5,\hfill \end{array}$$

(45)

we have the following Sylvester-like matrix equation:

$$\begin{array}{cc}\hfill & \stackrel{`}{A_1}{W}_2+U_3\stackrel{`}{B_1}+\stackrel{`}{A_2}{W}_1\stackrel{`}{B_2}+\stackrel{`}{A_3}{U}_1\stackrel{`}{B_3}+\stackrel{`}{A_4}Q\stackrel{`}{B_4}=\stackrel{`}{C_1}.\hfill \end{array}$$

(46)

Now, by applying Lemma 4, the Equation (46) has a solution if and only if

$$\begin{array}{cc}\hfill & R_{G_i}{L}_i=0,L_i{L}_{H_i}=0(i=\overline{1,4}),R_{\widehat{A_3}}E{L}_{\widehat{B_2}}=0.\hfill \end{array}$$

(47)

In this case, the expression of the general solution of (46) can be expressed as

$$\begin{array}{cc}\hfill & W_2={\stackrel{`}{A_1}}^{†}(\stackrel{`}{C_1}-\stackrel{`}{A_2}{W}_1\stackrel{`}{B_2}-\stackrel{`}{A_3}{U}_1\stackrel{`}{B_3}-\stackrel{`}{A_4}Q\stackrel{`}{B_4})-P_1\stackrel{`}{B_1}+L_{\stackrel{`}{A_1}}{P}_2,\hfill \\ \hfill & U_3=R_{\stackrel{`}{A_1}}(\stackrel{`}{C_1}-\stackrel{`}{A_2}{Z}_1\stackrel{`}{B_2}-\stackrel{`}{A_3}{Z}_2\stackrel{`}{B_3}-\stackrel{`}{A_4}{Z}_3\stackrel{`}{B_4}){\stackrel{`}{B_1}}^{†}+\stackrel{`}{A_1}{P}_1+P_3{R}_{\stackrel{`}{B_1}},\hfill \\ \hfill & W_1=A_{11}^{†}T{B}_{11}^{†}-A_{11}^{†}{A}_{22}{M}_1^{†}T{B}_{11}^{†}-A_{11}^{†}{S}_1{A}_{22}^{†}T{N}_1^{†}{B}_{22}{B}_{11}^{†}\hfill \\ \hfill & -A_{11}^{†}{S}_1{P}_4{R}_{N_1}{B}_{22}{B}_{11}^{†}+L_{A_{11}}{P}_5+P_6{R}_{B_{11}},\hfill \\ \hfill & U_1=M_1^{†}T{B}_{22}^{†}+S_1^{†}{S}_1{A}_{22}^{†}T{N}_1^{†}+L_{M_1}{L}_{S_1}{P}_7+P_8{R}_{B_{22}}+L_{M_1}{U}_4{R}_{N_1},\hfill \\ \hfill & Q=Q_{11}^0+L_{G_2}{J}_1+J_2{R}_{H_1}+L_{G_1}{J}_3{R}_{H_2},\hfill \\ \hfill & orQ=Q_{22}^0-L_{G_4}{K}_1-K_2{R}_{H_3}-L_{G_3}{K}_3{R}_{H_4},\hfill \end{array}$$

where

$$\begin{array}{cc}\hfill & \left(\begin{array}{c}{J}_1\\ K_1\end{array}\right)={\widehat{A_1}}^{†}(\overline{Q}-\widehat{C_1}{J}_3\widehat{D_1}-\widehat{C_2}{K}_3\widehat{D_2})-U_{11}\widehat{B_1}+L_{\widehat{A_1}}{U}_{12},\hfill \\ \hfill & \left(\begin{array}{cc}{J}_2& K_2\end{array}\right)=R_{\widehat{A_1}}(\overline{Q}-\widehat{C_1}{J}_3\widehat{D_1}-\widehat{C_2}{K}_3\widehat{D_2}){\widehat{B_1}}^{†}+\widehat{A_1}{U}_{11}+U_{21}{R}_{\widehat{B_1}},\hfill \\ \hfill & J_3={\widehat{A_2}}^{†}\overline{Q}{\widehat{B_2}}^{†}-{\widehat{A_2}}^{†}\widehat{A_3}{M}^{†}\overline{Q}{\widehat{B_2}}^{†}-{\widehat{A_2}}^{†}S{\widehat{A_3}}^{†}\overline{Q}{N}^{†}\widehat{B_3}{\widehat{B_2}}^{†}\hfill \\ \hfill & -{\widehat{A_2}}^{†}S{U}_{31}{R}_N\widehat{B_3}{\widehat{B_2}}^{†}+L_{\widehat{A_2}}{U}_{32}+U_{33}{R}_{\widehat{B_2}},\hfill \\ \hfill & K_3=M^{†}\overline{Q}{\widehat{B_3}}^{†}+S^{†}S{\widehat{A_3}}^{†}\overline{Q}{N}^{†}+L_M{L}_S{U}_{41}+U_{42}{R}_{\widehat{B_3}}+L_M{U}_{31}{R}_N,\hfill \end{array}$$

and $P_i$$(i=\overline{1,8})$, $U_{11}$, $U_{12}$, $U_{21}$, $U_{31}$, $U_{32}$, $U_{33}$, $U_{41}$ and $U_{42}$ are arbitrary matrices.

$\left(2\right)\iff \left(3\right)$: Under the conditions in (47), we have that the system (1) is solvable and hence there is a special solution $X_0$, $Y_0$, $Z_0$, $W_0$ and $V_0$ such that

$$\begin{array}{c}\hfill \begin{array}{c}\hfill \begin{array}{c}{A}_1{V}_0=C_1,V_0{B}_1=C_2,\\ A_3{X}_0+Y_0{B}_3=C_3,\\ A_2{Z}_0+Y_0{B}_2+A_5{V}_0{B}_5=C_5,\\ A_4{Z}_0+W_0{B}_4=C_4.\end{array}\end{array}\end{array}$$

(48)

By using Lemma 1, it is easy to check that:

$$\begin{array}{cc}\hfill & R_{A_1}{C}_1=0\iff r\left(\begin{array}{cc}{C}_1& A_1\end{array}\right)=r\left(A_1\right),C_2{L}_{B_1}=0\iff r\left(\begin{array}{c}{C}_2\\ B_1\end{array}\right)=r\left(B_1\right),\hfill \\ \hfill & R_{A_i}{C}_i{L}_{B_i}=0\iff r\left(\begin{array}{cc}{C}_i& A_i\\ B_i& 0\end{array}\right)=r\left(\begin{array}{c}{A}_i\end{array}\right)+r\left(\begin{array}{c}{B}_i\end{array}\right),(i=3,4).\hfill \end{array}$$

By Lemma 4, it suffices to show that the conditions (13)–(21) are equivalent to the conditions (26)–(34), respectively.

$$\begin{array}{cc}\hfill & (13)\stackrel{Equation\left(22\right)}{\iff }\hfill \\ \hfill & r\left(\begin{array}{ccccc}\stackrel{`}{C_1}& A_2& A_3& A_5{L}_{A_1}& A_2{L}_{A_4}\\ R_{B_3}{B}_2& 0& 0& 0& 0\end{array}\right)=r\left(\begin{array}{c}{R}_{B_3}{B}_2\end{array}\right)+r\left(\begin{array}{cccc}{A}_2& A_3& A_5{L}_{A_1}& A_2{L}_{A_4}\end{array}\right)\hfill \\ \hfill & \underset{operations,Equation\left(48\right)}{\overset{Elementaryrows\left(columns\right)}{\iff }}\hfill \\ \hfill & r\left(\begin{array}{ccccc}{C}_5& A_2& A_4& A_5& 0\\ C_1{B}_5& 0& 0& A_1& 0\\ B_2& 0& 0& 0& B_4\end{array}\right)=r\left(\begin{array}{cc}{B}_2& B_4\end{array}\right)+r\left(\begin{array}{ccc}{A}_5& A_2& A_4\\ A_1& 0& 0\end{array}\right)\iff \left(26\right).\hfill \\ \hfill & (14)\stackrel{Equation\left(22\right)}{\iff }\hfill \\ \hfill & r\left(\begin{array}{cc}\stackrel{`}{C_1}& A_2{A}_3\\ R_{B_3}& 0\\ B_2& 0\\ R_{B_1}{B}_5& 0\\ R_{B_4}{B}_2& 0\end{array}\right)=r\left(A_2{A}_3\right)+r\left(\begin{array}{c}{R}_{B_3}\\ B_2\\ R_{B_1}{B}_5\\ R_{B_4}{B}_2\end{array}\right)\hfill \\ \hfill & \underset{operations,Equation\left(48\right)}{\overset{Elementaryrows\left(columns\right)}{\iff }}\hfill \\ \hfill & r\left(\begin{array}{cccc}{C}_5& A_2{A}_3& A_2{C}_3& A_5{C}_2\\ I& 0& B_3& 0\\ B_2& 0& 0& 0\\ B_5& 0& 0& B_1\end{array}\right)=r\left(A_2{A}_3\right)+r\left(\begin{array}{ccc}I& B_3& 0\\ B_2& 0& 0\\ B_5& 0& B_1\end{array}\right)\iff \left(27\right).\hfill \end{array}$$

$$\begin{array}{cc}\hfill & (15)\stackrel{Equation\left(22\right)}{\iff }\hfill \\ \hfill & r\left(\begin{array}{cccc}\stackrel{`}{C_1}& A_2& A_4& A_2{A}_3\\ R_{B_1}{B}_5& 0& 0& 0\\ R_{B_4}{B}_2& 0& 0& 0\end{array}\right)=r\left(\begin{array}{c}{R}_{B_1}{B}_5\\ R_{B_4}{B}_2\end{array}\right)+r\left(\begin{array}{ccc}{A}_2& A_4& A_2{A}_3\end{array}\right)\hfill \\ \hfill & \underset{operations,Equation\left(48\right)}{\overset{Elementaryrows\left(columns\right)}{\iff }}\hfill \\ \hfill & r\left(\begin{array}{ccccc}{C}_5& A_2& A_4& A_5{C}_2& C_4\\ B_5& 0& 0& B_1& 0\\ B_2& 0& 0& 0& B_4\end{array}\right)=r\left(\begin{array}{ccc}{B}_5& B_1& 0\\ B_2& 0& B_4\end{array}\right)+r\left(\begin{array}{cc}{A}_2& A_4\end{array}\right)\iff \left(28\right).\hfill \\ \hfill & (16)\stackrel{Equation\left(22\right)}{\iff }\hfill \\ \hfill & r\left(\begin{array}{cccc}\stackrel{`}{C_1}& A_2& A_5{L}_{A_1}& A_2{A}_3\\ B_2& 0& 0& 0\\ R_{B_4}{B}_2& 0& 0& 0\end{array}\right)=r\left(\begin{array}{c}{B}_2\\ R_{B_4}{B}_2\end{array}\right)+r\left(\begin{array}{ccc}{A}_2& A_5{L}_{A_1}& A_2{A}_3\end{array}\right)\hfill \\ \hfill & \underset{operations,Equation\left(48\right)}{\overset{Elementaryrows\left(columns\right)}{\iff }}\left(29\right).\hfill \\ \hfill & (17)\stackrel{Equation\left(22\right)}{\iff }\hfill \\ \hfill & r\left(\begin{array}{cccc}\stackrel{`}{C_1}& A_4& A_5{L}_{A_1}& A_2{A}_3\\ R_{B_3}& 0& 0& 0\\ R_{B_4}{B}_2& 0& 0& 0\end{array}\right)=r\left(\begin{array}{c}{R}_{B_3}\\ R_{B_4}{B}_2\end{array}\right)+r\left(\begin{array}{ccc}{A}_4& A_5{L}_{A_1}& A_2{A}_3\end{array}\right)\hfill \\ \hfill & \underset{operations,Equation\left(48\right)}{\overset{Elementaryrows\left(columns\right)}{\iff }}\hfill \\ \hfill & r\left(\begin{array}{cccccc}{C}_5& A_4& A_5& A_2{A}_3& A_2{C}_3& C_4\\ I& 0& 0& 0& B_3& 0\\ B_2& 0& 0& 0& 0& B_4\\ C_1{B}_5& 0& A_1& 0& 0& 0\end{array}\right)=r\left(\begin{array}{ccc}I& B_3& 0\\ B_2& 0& B_4\end{array}\right)+r\left(\begin{array}{ccc}{A}_5& A_2{A}_3& A_4\\ A_1& 0& 0\end{array}\right)\iff \left(30\right).\hfill \\ \hfill & (18)Equation\left(22\right)\hfill \\ \hfill & r\left(\begin{array}{ccc}\stackrel{`}{C_1}& A_5{L}_{A_1}& A_2{A}_3\\ R_{B_3}& 0& 0\\ B_2& 0& 0\\ R_{B_4}{B}_2& 0& 0\end{array}\right)=r\left(\begin{array}{c}{R}_{B_3}\\ B_2\\ R_{B_4}{B}_2\end{array}\right)+r\left(\begin{array}{cc}{A}_5{L}_{A_1}& A_2{A}_3\end{array}\right)\hfill \\ \hfill & \underset{operations,Equation\left(48\right)}{\overset{Elementaryrows\left(columns\right)}{\iff }}\hfill \\ \hfill & r\left(\begin{array}{cccc}{C}_5& A_5& A_2{A}_3& A_2{C}_3\\ I& 0& 0& B_3\\ B_2& 0& 0& 0\\ C_1{B}_5& A_1& 0& 0\end{array}\right)=r\left(\begin{array}{cc}I& B_3\\ B_2& 0\end{array}\right)+r\left(\begin{array}{cc}{A}_5& A_2{A}_3\\ A_1& 0\end{array}\right)\iff \left(31\right).\hfill \\ \hfill & (19)Equation\left(22\right)\hfill \\ \hfill & r\left(\begin{array}{ccc}\stackrel{`}{C_1}& A_4& A_2{A}_3\\ R_{B_3}& 0& 0\\ R_{B_1}{B}_5& 0& 0\\ R_{B_4}{B}_2& 0& 0\end{array}\right)=r\left(\begin{array}{c}{R}_{B_3}\\ R_{B_1}{B}_5\\ R_{B_4}{B}_2\end{array}\right)+r\left(\begin{array}{cc}{A}_4& A_2{A}_3\end{array}\right)\hfill \\ \hfill & \underset{operations,Equation\left(48\right)}{\overset{Elementaryrows\left(columns\right)}{\iff }}\hfill \\ \hfill & r\left(\begin{array}{cccccc}{C}_5& A_4& A_2{A}_3& A_5{C}_2& C_4& A_2{C}_3\\ I& 0& 0& 0& 0& B_3\\ B_5& 0& 0& B_1& 0& 0\\ B_2& 0& 0& 0& B_4& 0\end{array}\right)=r\left(\begin{array}{cccc}I& B_3& 0& 0\\ B_5& 0& B_1& 0\\ B_2& 0& 0& B_4\end{array}\right)+r\left(\begin{array}{cc}{A}_4& A_2{A}_3\end{array}\right)\iff \left(32\right).\hfill \end{array}$$

$$\begin{array}{cc}\hfill & (20)\stackrel{Equation\left(22\right)}{\iff }\hfill \\ \hfill & r\left(\begin{array}{ccc}\stackrel{`}{C_1}& A_2& A_2{A}_3\\ B_2& 0& 0\\ R_{B_1}{B}_5& 0& 0\\ R_{B_4}{B}_2& 0& 0\end{array}\right)=r\left(\begin{array}{c}{B}_2\\ R_{B_1}{B}_5\\ R_{B_4}{B}_2\end{array}\right)+r\left(\begin{array}{cc}{A}_2& A_2{A}_3\end{array}\right)\hfill \\ \hfill & \underset{operations,Equation\left(48\right)}{\overset{Elementaryrows\left(columns\right)}{\iff }}\left(33\right).\hfill \\ \hfill & (21)\stackrel{Equation\left(22\right)}{\iff }\hfill \\ \hfill & r\left(\begin{array}{ccccccc}\stackrel{`}{C_1}& A_2& A_2{A}_3& 0& 0& 0& A_5{L}_{A_1}\\ B_2& 0& 0& 0& 0& 0& 0\\ R_{B_4}{B}_2& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& -\stackrel{`}{C_1}& A_4& A_2{A}_3& A_5{L}_{A_1}\\ 0& 0& 0& R_{B_3}& 0& 0& 0\\ 0& 0& 0& R_{B_4}{B}_2& 0& 0& 0\\ R_{B_1}{B}_5& 0& 0& R_{B_1}{B}_5& 0& 0& 0\end{array}\right)\hfill \\ \hfill & =r\left(\begin{array}{cc}{B}_2& 0\\ R_{B_4}{B}_2& 0\\ 0& R_{B_3}\\ 0& R_{B_4}{B}_2\\ R_{B_1}{B}_5& R_{B_1}{B}_5\end{array}\right)+r\left(\begin{array}{ccccc}{A}_2& A_2{A}_3& 0& 0& A_5{L}_{A_1}\\ 0& 0& A_4& A_2{A}_3& A_5{L}_{A_1}\end{array}\right)\hfill \\ \hfill & \underset{operations,Equation\left(48\right)}{\overset{Elementaryrows\left(columns\right)}{\iff }}\hfill \\ \hfill & r\left(\begin{array}{ccccccccc}{C}_5& A_2& 0& 0& 0& A_5& 0& 0& 0\\ B_2& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& -C_5& A_4& A_2{A}_3& A_5& -A_2{C}_3& -C_4& -A_5{C}_2\\ 0& 0& I& 0& 0& 0& B_3& 0& 0\\ 0& 0& B_2& 0& 0& 0& 0& B_4& 0\\ B_5& 0& B_5& 0& 0& 0& 0& 0& B_1\\ C_1{B}_5& 0& 0& 0& 0& A_1& 0& 0& 0\end{array}\right)\hfill \\ \hfill & =r\left(\begin{array}{ccccc}{B}_2& 0& 0& 0& 0\\ 0& I& B_3& 0& 0\\ 0& B_2& 0& B_4& 0\\ B_5& B_5& 0& 0& B_1\end{array}\right)+r\left(\begin{array}{cccc}{A}_2& 0& 0& A_5\\ 0& A_4& A_2{A}_3& A_5\\ 0& 0& 0& A_1\end{array}\right)\iff \left(34\right).\hfill \end{array}$$

   □

3.1. Some Special Cases to the System (1)

In this section, we consider some special cases of the system (1). The following corollary coincides with one of the essential findings in [33].

Corollary 1.

Let $A_1$, $B_1$, $C_1$, $D_1$ and $E_1$ be given. Set

$$\begin{array}{cc}\hfill & A_3=R_{A_1}{C}_1,B_3=D_1{L}_{B_1},C_3=R_{A_1}{E}_1{L}_{B_1}.\hfill \end{array}$$

Then the following statements are equivalent:

(1) The matrix Equation (8) is solvable.

(2)

$$\begin{array}{cc}\hfill & R_{A_3}{C}_3=0,C_3{L}_{B_3}=0.\hfill \end{array}$$

(3)

$$\begin{array}{cc}\hfill & r\left(\begin{array}{ccc}{E}_1& C_1& A_1\\ B_1& 0& 0\end{array}\right)=r\left(\begin{array}{cc}{C}_1& A_1\end{array}\right)+r\left(\begin{array}{c}{B}_1\end{array}\right),r\left(\begin{array}{cc}{E}_1& A_1\\ D_1& 0\\ B_1& 0\end{array}\right)=r\left(\begin{array}{c}{A}_1\end{array}\right)+r\left(\begin{array}{cc}{D}_1& B_1\end{array}\right).\hfill \end{array}$$

In this case, the general solution of (8) can be expressed as

$$\begin{array}{cc}\hfill & X=A_1^{†}(E_1-C_{1}Z{D}_1)+W_1{B}_1+L_{A_1}{W}_2,\hfill \\ \hfill & Y=R_{A_1}(E_1-C_{1}Z{D}_1)B_1^{†}+A_1{W}_1+W_3{R}_{B_1},\hfill \\ \hfill & Z=A_3^{†}{C}_3{B}_3^{†}+L_{A_3}U+V{R}_{B_3},\hfill \end{array}$$

where $W_1,$ $W_2,$ $W_3,$ U and V are arbitrary matrices over the $\mathbb{H}$ with suitable sizes.

The following corollary is matching with the main result in [30].

Corollary 2.

Let $A_i$, $B_i$ and $C_i$$(i=\overline{1,3})$ be given. Set

$$\begin{array}{cc}\hfill & A=R_{\left(A_2{A}_1\right)}{A}_2,B=R_{B_1}{L}_{\left(R_{B_3}{B}_2\right)},C=R_{\left(A_2{A}_1\right)}{A}_3,D=B_2{L}_{\left(R_{B_3}{B}_2\right)},\hfill \\ & C_4=C_2-A_2{R}_{A_1}{C}_1{B}_1^{†}-R_{A_3}{C}_3{B}_3^{†}{B}_2,E=R_{\left(A_2{A}_1\right)}{C}_4{L}_{\left(R_{B_3}{B}_2\right)},\hfill \\ \hfill & M=R_{A}C,N=D{L}_B,S=C{L}_M.\hfill \end{array}$$

Then the following statements are equivalent:

(1) The system

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}\hfill & A_{1}X+Y{B}_1=C_1,\hfill \\ \hfill & A_{2}Y+Z{B}_2=C_2,\hfill \\ \hfill & A_{3}W+Z{B}_3=C_3,\hfill \end{array}\right.\end{array}$$

(49)

 is solvable.

(2)

$$\begin{array}{cc}\hfill & R_{A_i}{C}_i{L}_{B_i}=0,(i=\overline{1,3}),R_M{R}_{A}E=0,E{L}_B{L}_N=0,R_{A}E{L}_D=0,R_{C}E{L}_B=0.\hfill \end{array}$$

(3)

$$\begin{array}{cc}\hfill & r\left(\begin{array}{cc}{C}_i& A_i\\ B_i& 0\end{array}\right)=r\left(\begin{array}{c}{A}_i\end{array}\right)+r\left(\begin{array}{c}{B}_i\end{array}\right)(i=\overline{1,3}),r\left(\begin{array}{cccc}{C}_2& C_3& A_2& A_3\\ B_2& B_3& 0& 0\end{array}\right)=r\left(\begin{array}{cc}{A}_2& A_3\end{array}\right)+r\left(\begin{array}{cc}{B}_2& B_3\end{array}\right),\hfill \\ \hfill & r\left(\begin{array}{cccc}{C}_2{B}_1-A_2{C}_1& A_2{A}_1& A_3& C_3\\ B_2{B}_1& 0& 0& B_3\end{array}\right)=r\left(\begin{array}{cc}{A}_2{A}_1& A_3\end{array}\right)+r\left(\begin{array}{cc}{B}_2{B}_1& B_3\end{array}\right),\hfill \\ \hfill & r\left(\begin{array}{cc}{C}_2{B}_1-A_2{C}_1& A_2{A}_1\\ B_2{B}_1& 0\end{array}\right)=r\left(\begin{array}{c}{A}_2{A}_1\end{array}\right)+r\left(\begin{array}{c}{B}_2{B}_1\end{array}\right).\hfill \end{array}$$

In this case, the general solution of (49) can be expressed as

$$\begin{array}{cc}\hfill & X=A_1^{†}{C}_1+U_1{B}_1+L_{A_1}{U}_2,Y=R_{A_1}{C}_1{B}_1^{†}+A_1{U}_1+U_3{R}_{B_1},\hfill \\ \hfill & Z=R_{A_3}{C}_3{B}_3^{†}+A_3{V}_1+V_3{R}_{B_3},W=A_3^{†}{C}_3+V_1{B}_3+L_{A_3}{V}_2,\hfill \\ \hfill & U_1={\left(A_2{A}_1\right)}^{†}(C_4-A_2{U}_3{R}_{B_1}-A_3{V}_1{B}_2)+T_7\left(R_{B_3}{B}_2\right)+L_{\left(A_2{A}_1\right)}{T}_6,\hfill \\ \hfill & V_3=R_{\left(A_2{A}_1\right)}(C_4-A_2{U}_3{R}_{B_1}-A_3{V}_1{B}_2){\left(R_{B_3}{B}_2\right)}^{†}+\left(A_2{A}_1\right)T_7+T_8{R}_{\left(R_{B_3}{B}_2\right)},\hfill \\ \hfill & U_3=A^{†}E{B}^{†}-A^{†}C{M}^{†}E{B}^{†}-A^{†}S{C}^{†}E{N}^{†}D{B}^{†}-A^{†}S{T}_2{R}_{N}D{B}^{†}+L_A{T}_4+T_5{R}_B,\hfill \\ \hfill & V_1=M^{†}E{D}^{†}+S^{†}S{C}^{†}E{N}^{†}+L_M{L}_S{T}_1+L_M{T}_2{R}_N+T_3{R}_D,\hfill \end{array}$$

where $U_2$, $V_2$, $T_1,\cdots ,T_8$ are arbitrary matrices over $\mathbb{H}$ with suitable sizes.

The following corollary can be considered as an essential finding in [11,29,34].

Corollary 3.

Let $A_i$, $B_i$ and $C_i$$(i=1,2)$ be given. Set

$$\begin{array}{cc}\hfill & D_1=R_{B_1}{B}_2,A=R_{A_2}{A}_1,B=B_2{L}_{D_1},C=R_{A_2}(R_{A_1}{C}_1{B}_1^{†}{B}_2+C_2)L_{D_1}\hfill \end{array}$$

Then the following statements are equivalent:

(1) The system (5) is solvable.

(2)

$$\begin{array}{cc}\hfill & R_{A_1}{C}_1{L}_{B_1}=0,R_{A}C=0,C{L}_B=0.\hfill \end{array}$$

(3)

$$\begin{array}{cc}\hfill & r\left(\begin{array}{cc}{C}_i& A_i\\ B_i& 0\end{array}\right)=r\left(\begin{array}{c}{A}_i\end{array}\right)+r\left(\begin{array}{c}{B}_i\end{array}\right)(i=1,2),\hfill \\ \hfill & r\left(\begin{array}{cccc}{C}_2& C_1& A_1& A_2\\ B_2& B_1& 0& 0\end{array}\right)=r\left(\begin{array}{cc}{A}_1& A_2\end{array}\right)+r\left(\begin{array}{cc}{B}_1& B_2\end{array}\right).\hfill \end{array}$$

In this case, the general solution of (5) can be expressed as

$$\begin{array}{cc}\hfill & X=A_1^{†}{C}_1+U_1{B}_1+L_{A_1}{W}_1,\hfill \\ \hfill & Y=R_{A_1}{C}_1{B}_1^{†}+A_1{U}_1+V_1{R}_{B_1},\hfill \\ \hfill & Z=A_2^{†}(C_2+R_{A_1}{C}_1{B}_1^{†}{B}_2+A_1{U}_1{B}_2)+W_4{D}_1+L_{A_2}{W}_6,\hfill \\ \hfill & U_1=A^{†}C{B}^{†}+L_A{W}_2+W_3{R}_B,\hfill \\ \hfill & V_1=R_{A_2}(C_2{R}_{A_1}{C}_1{B}_1^{†}{B}_2+A_1{U}_1{B}_2)D_1^{†}+A_2{W}_4+W_5{R}_{D_1},\hfill \end{array}$$

where $W_1,\cdots ,W_6$ are arbitrary matrices with suitable sizes over $\mathbb{H}$.

3.2. Algorithm with a Numerical Example

In this subsection, we carry out an algorithm(see Algorithm 1) and a numerical example to justify our main findings. The numerical calculations can be done using MATLAB 2018 (R2018b).

Algorithm 1 Calculate the general solution to the system (1).

1. Input the system (1) with coefficients $A_k$, $B_k$ and $C_k,$ $(k=\overline{1,5})$ with viable dimensions over $\mathbb{H}$. 2. Compute all matrices that appeared in (12) and (22). 3. Check whether the Moore–Penrose inverses conditions in Theorem 1 are satisfied or not. If not, return “The system (1) is unsolvable”; 4. Else compute X, Y, Z, W and V by (35)–(39). 5. Output the general solution of the system (1) is (X, Y, Z, W, V).  Note that the conditions in $\left(3\right)$ can be exchange by rank equalities conditions (24)–(34).

Example 1.

Consider the system of Sylvester-like quaternion matrix (1), where

$$\begin{array}{cc}\hfill & A_1=\left(\begin{array}{ccc}\mathbf{i}& \mathbf{0}& \mathbf{j}\end{array}\right),A_2=\left(\begin{array}{ccc}\mathbf{i}& \mathbf{0}& -\mathbf{k}\\ \mathbf{0}& \mathbf{0}& \mathbf{0}\end{array}\right),A_3=\left(\begin{array}{cc}\mathbf{0}& \mathbf{1}\\ -\mathbf{i}& \mathbf{0}\\ \mathbf{3}& \mathbf{0}\end{array}\right),A_4=\left(\begin{array}{cc}\mathbf{0}& \mathbf{0}\\ \mathbf{0}& \mathbf{k}\end{array}\right),\hfill \\ \hfill & A_5=\left(\begin{array}{ccc}\mathbf{i}& \mathbf{0}& \mathbf{0}\\ \mathbf{0}& \mathbf{2}& \mathbf{k}\end{array}\right),B_1=\left(\begin{array}{c}-\mathbf{j}\\ \mathbf{0}\\ -\mathbf{k}\end{array}\right),B_2=\left(\begin{array}{cc}\mathbf{0}& \mathbf{0}\\ \mathbf{0}& -\mathbf{j}\\ \mathbf{0}& \mathbf{0}\\ \mathbf{k}& \mathbf{0}\end{array}\right),B_3=\left(\begin{array}{c}\mathbf{1}-\mathbf{i}\\ \mathbf{0}\end{array}\right),\hfill \end{array}$$

$$\begin{array}{cc}\hfill & B_4=\left(\begin{array}{c}\mathbf{i}\\ \mathbf{1}+\mathbf{k}\\ -\mathbf{1}\\ \mathbf{0}\end{array}\right),B_5=\left(\begin{array}{cc}\mathbf{0}& -\mathbf{j}\\ \mathbf{1}& \mathbf{0}\\ \mathbf{k}& -\mathbf{k}\end{array}\right),C_1=\left(\begin{array}{ccc}\mathbf{0}& -\mathbf{i}& \mathbf{0}\end{array}\right),C_2=\left(\begin{array}{c}\mathbf{0}\\ \mathbf{k}\\ \mathbf{0}\end{array}\right),\hfill \\ \hfill & C_3=\left(\begin{array}{c}\mathbf{2}\mathbf{j}+\mathbf{k}\\ \mathbf{1}\\ \mathbf{3}\mathbf{i}\end{array}\right),C_4=\left(\begin{array}{c}\mathbf{i}+\mathbf{j}\\ \mathbf{2}\mathbf{i}-\mathbf{j}+\mathbf{k}\end{array}\right),C_5=\left(\begin{array}{cc}-\mathbf{i}+\mathbf{k}& -\mathbf{k}\\ \mathbf{1}& \mathbf{k}\end{array}\right).\hfill \end{array}$$

Straightforward calculations in Algorithm 1, using the quaternion package on MatLab or Maple software, it can be found that these matrices obey all the equalities in (23). Upon computation, we observe that $A_{11}=0,$ $B_{11}=0,$ $B_{22}=0,$ $B_{33}=0,$ $T_1=0,$ $N_1=0,$ $S_1=0,$ $Q=0,$ $H_i=0,$ $L_i=0$ $(i=\overline{1,4}),$ $G_1=0,$ $G_3=0,$ $G_2=G_4=A_{33},$ ${\widehat{B}}_1=\left(\begin{array}{c}{I}_3\\ I_3\end{array}\right),$ ${\widehat{C}}_1={\widehat{D}}_1={\widehat{C}}_2={\widehat{D}}_2=I_3,$ ${\widehat{A}}_2={\widehat{A}}_3=R_{{\widehat{A}}_1}$ and ${\widehat{B}}_2={\widehat{B}}_3=L_{{\widehat{B}}_1},$ where $I_j$ is the identity matrix of order j. Consequently, the system (1) is solvable. Hence the general solution to (1) is

$$\begin{array}{cc}\hfill & X=X_0+P_1{M}_{11}+M_{12}{P}_2{M}_{13},\hfill \\ \hfill & Y=Y_0+M_{21}{P}_1{M}_{22}+M_{23}{P}_2+M_{24}{P}_5{M}_{25}+P_6{M}_{26},\hfill \\ \hfill & W=W_0+M_{31}{P}_{11}{M}_{32}+P_8{M}_{33}+M_{34}{W}_2,\hfill \\ \hfill & Z=Z_0+M_{41}{P}_8+M_{42}{P}_1{M}_{43}+P_3{M}_{44},\hfill \\ \hfill & V=V_0+M_{51}{U}_{32}{M}_{52},\hfill \end{array}$$

where $P_1,$ $P_2,$ $P_5,$ $P_6,$ $P_{11}=P_7+U_4,$ $P_8,$ $W_2,$ $P_3$ and $U_{32}$ are arbitrary matrices over $\mathbb{H}$, and

$$\begin{array}{cc}\hfill & X_0=\left(\begin{array}{c}\mathbf{i}\\ \mathbf{j}\end{array}\right),Y_0=\left(\begin{array}{cc}\mathbf{j}& \mathbf{0}\\ \mathbf{0}& -\mathbf{k}\\ \mathbf{0}& \mathbf{1}\end{array}\right),Z_0=\left(\begin{array}{cccc}\mathbf{1}+\mathbf{k}& \mathbf{0}& \mathbf{0}& \mathbf{0}\\ \mathbf{0}& \mathbf{i}& \mathbf{0}& -\mathbf{k}\end{array}\right),W_0=\left(\begin{array}{c}-\mathbf{1}\\ \mathbf{1}-\mathbf{j}\end{array}\right),\hfill \\ \hfill & V_0=\left(\begin{array}{ccc}\mathbf{0}& -\mathbf{1}& \mathbf{0}\\ -\mathbf{i}& \mathbf{0}& \mathbf{0}\\ \mathbf{0}& \mathbf{0}& \mathbf{0}\end{array}\right),M_{11}=-{\stackrel{`}{B}}_1{B}_3=\left(\begin{array}{c}\mathbf{0}\\ \mathbf{0}\\ \mathbf{0}\\ \mathbf{j}-\mathbf{k}\end{array}\right),M_{12}=L_{{\stackrel{`}{A}}_1}=\left(\begin{array}{cc}\mathbf{0.1000}& -\mathbf{0.3000}\mathbf{j}\\ \mathbf{0.3000}\mathbf{j}& \mathbf{0.9000}\end{array}\right),\hfill \\ \hfill & M_{13}=B_3,M_{21}=-A_3,M_{22}={\stackrel{`}{B}}_1=\left(\begin{array}{cc}\mathbf{0}\hfill & -\mathbf{0.2500}+\mathbf{0.2500}\mathbf{k}\\ \mathbf{0}\hfill & -\mathbf{0.5000}\mathbf{j}\\ \mathbf{0}\hfill & -\mathbf{0.2500}\mathbf{i}-\mathbf{0.2500}\mathbf{j}\\ \mathbf{1.000}\mathbf{k}& \mathbf{0}\end{array}\right),\hfill \\ \hfill & M_{25}=R_{B_3}=\left(\begin{array}{cc}\mathbf{0}& \mathbf{0}\\ \mathbf{0}& \mathbf{1.000}\end{array}\right),M_{23}=A_3{L}_{{\stackrel{`}{A}}_1}=\left(\begin{array}{cc}\mathbf{0.3000}\mathbf{j}& \mathbf{0.9000}\\ -\mathbf{0.1000}\mathbf{i}& \mathbf{0.3000}\mathbf{k}\\ \mathbf{0.3000}& -\mathbf{0.9000}\mathbf{j}\end{array}\right),\hfill \\ \hfill & M_{24}=L_{A_{11}}=\left(\begin{array}{ccc}\mathbf{0.5000}& \mathbf{0}& \mathbf{0.5000}\mathbf{j}\\ \mathbf{0}& \mathbf{1.0000}& \mathbf{0}\\ -\mathbf{0.5000}\mathbf{j}& \mathbf{0}& \mathbf{0.5000}\end{array}\right),M_{26}=R_{B_{11}}{R}_{B_3}=\left(\begin{array}{cc}\mathbf{0}& \mathbf{0}\\ \mathbf{0}& \mathbf{0.6667}+\mathbf{0.4714}\mathbf{j}\end{array}\right),\hfill \\ \hfill & M_{31}=M_{34}=L_{M_1}=L_{A_4}=\left(\begin{array}{cc}\mathbf{1}& \mathbf{0}\\ \mathbf{0}& \mathbf{0}\end{array}\right),M_{32}=M_{33}=B_4,M_{41}=A_4,M_{42}={\stackrel{`}{A}}_1,\hfill \\ \hfill & M_{43}=\left(\begin{array}{cccc}\mathbf{0.7500}& -\mathbf{0.2500}\mathbf{i}-\mathbf{0.2500}\mathbf{j}& \mathbf{0.2500}\mathbf{i}& \mathbf{0}\\ \mathbf{0.2500}\mathbf{i}+\mathbf{0.2500}\mathbf{j}& \mathbf{0.5000}& \mathbf{0.2500}+\mathbf{0.2500}\mathbf{k}& \mathbf{0}\\ -\mathbf{0.2500}\mathbf{i}& \mathbf{0.2500}& \mathbf{0.7500}-\mathbf{0.2500}\mathbf{k}& \mathbf{0}\\ \mathbf{0}& \mathbf{0}& \mathbf{0}& \mathbf{1.000}\end{array}\right),\hfill \\ \hfill & M_{44}=R_{{\stackrel{`}{B}}_1}{R}_{B_4}=\left(\begin{array}{cccc}\mathbf{0.5000}& \mathbf{0}& \mathbf{0.5000}\mathbf{i}& \mathbf{0}\\ \mathbf{0}& \mathbf{0}& \mathbf{0}& \mathbf{0}\\ -\mathbf{0.5000}\mathbf{i}& \mathbf{0}& \mathbf{0.5000}& \mathbf{0}\\ \mathbf{0}& \mathbf{0}& \mathbf{0}& \mathbf{0}\end{array}\right),M_{51}=L_{A_1}{L}_{{\widehat{A}}_2}=\left(\begin{array}{ccc}\mathbf{0.5000}& \mathbf{0}& \mathbf{0.5000}\mathbf{k}\\ \mathbf{0}& \mathbf{0}& \mathbf{0}\\ \mathbf{0}& \mathbf{0}& \mathbf{0.5000}\end{array}\right),\hfill \end{array}$$

$$\begin{array}{cc}\hfill & M_{52}=R_{{\widehat{B}}_2}{R}_{B_1}=\left(\begin{array}{ccc}\mathbf{0.5000}& \mathbf{0}& \mathbf{0.5000}\mathbf{i}\\ \mathbf{0}& \mathbf{1.0000}& \mathbf{0}\\ -\mathbf{0.5000}\mathbf{i}& \mathbf{0}& \mathbf{0.5000}\end{array}\right).\hfill \end{array}$$

4. The Solvability Conditions and the General Solutions to the Systems (2) and (3)

We can easily investigate systems (2) and (3) by the same method and techniques we used in investigating the system (1). We merely present the main results of (2) and (3) in this part, omitting the intricacies of their proofs.

Let $A_i,$ $B_i$ and $C_i$$(i=\overline{1,5})$ be given. For simplicity, consider (12) such that

$$\begin{array}{cc}\hfill & \stackrel{`}{A_1}=A_2{L}_{A_4},\stackrel{`}{B_1}=R_{B_3}{B}_2,\stackrel{`}{A_2}=A_2,\stackrel{`}{B_2}=B_4,\stackrel{`}{A_3}=A_3,\stackrel{`}{B_3}=B_2,\stackrel{`}{A_4}=A_5{L}_{A_1},\hfill \\ \hfill & \stackrel{`}{B_4}=R_{B_1}{B}_5,\stackrel{`}{C_1}=C_5-[A_2{A}_4^{†}{C}_4+R_{A_3}{C}_3{B}_3^{†}{B}_2+A_5(A_1^{†}{C}_1+L_{A_1}{C}_2{B}_1^{†})B_5].\hfill \end{array}$$

(50)

Theorem 2.

The following statements are equivalent:

(1) The system (2) is solvable.

(2)

$$\begin{array}{cc}\hfill & R_{A_1}{C}_1=0,C_2{L}_{B_1}=0,A_1{C}_2=C_1{B}_1,R_{A_i}{C}_i{L}_{B_i}=0(i=3,4),\hfill \\ \hfill & R_{G_i}{L}_i=0,L_i{L}_{H_i}=0(i=\overline{1,4}),R_{\widehat{A_3}}E{L}_{\widehat{B_2}}=0.\hfill \end{array}$$

(51)

(3)

$$\begin{array}{cc}\hfill & r\left(\begin{array}{cc}{C}_1& A_1\end{array}\right)=r\left(A_1\right),r\left(\begin{array}{c}{C}_2\\ B_1\end{array}\right)=r\left(B_1\right),A_1{C}_2=C_1{B}_1,\hfill \\ \hfill & r\left(\begin{array}{cc}{C}_i& A_i\\ B_i& 0\end{array}\right)=r\left(\begin{array}{c}{A}_i\end{array}\right)+r\left(\begin{array}{c}{B}_i\end{array}\right)(i=3,4),\hfill \\ \hfill & r\left(\begin{array}{ccccc}{C}_5& C_3& A_2& A_3& A_5\\ B_2& B_3& 0& 0& 0\\ 0& 0& C_1{B}_5& 0& A_1\end{array}\right)=r\left(\begin{array}{cc}{B}_2& B_3\end{array}\right)+r\left(\begin{array}{ccc}{A}_2& A_3& A_5\\ 0& 0& A_1\end{array}\right),\hfill \\ \hfill & r\left(\begin{array}{ccccc}{C}_5& A_5{C}_2& C_3& A_2& A_3\\ B_5& B_1& 0& 0& 0\\ B_2& 0& B_3& 0& 0\end{array}\right)=r\left(\begin{array}{cc}{A}_2& A_3\end{array}\right)+r\left(\begin{array}{ccc}{B}_5& B_1& 0\\ B_2& 0& B_3\end{array}\right),\hfill \\ \hfill & r\left(\begin{array}{ccc}{C}_5& A_5{C}_2& A_2\\ B_4& 0& 0\\ B_2& 0& 0\\ B_5& B_1& 0\\ C_4& 0& A_4\end{array}\right)=r\left(\begin{array}{cc}{B}_4& 0\\ B_2& 0\\ B_5& B_1\end{array}\right)+r\left(\begin{array}{c}{A}_2\\ A_4\end{array}\right),\hfill \\ \hfill & r\left(\begin{array}{ccc}{C}_5& A_2& A_5\\ B_2& 0& 0\\ C_1{B}_5& 0& A_1\end{array}\right)=r\left(\begin{array}{cc}{A}_2& A_5\\ 0& A_1\end{array}\right)+r\left(\begin{array}{c}{B}_2\end{array}\right),\hfill \\ \hfill & r\left(\begin{array}{ccccc}{C}_5& 0& A_3& A_5& A_2\\ B_4& 0& 0& 0& 0\\ B_2& B_3& 0& 0& 0\\ C_1{B}_5& 0& 0& A_1& 0\\ C_4& 0& 0& 0& A_4\end{array}\right)=r\left(\begin{array}{ccc}{A}_3& A_5& A_2\\ 0& A_1& 0\\ 0& 0& A_4\end{array}\right)+r\left(\begin{array}{cc}{B}_4& 0\\ B_2& B_3\end{array}\right),\hfill \\ \hfill & r\left(\begin{array}{ccc}{C}_5& A_5& A_2\\ B_4& 0& 0\\ B_2& 0& 0\\ C_1{B}_5& A_1& 0\\ C_4& 0& A_4\end{array}\right)=r\left(\begin{array}{c}{B}_4\\ B_2\end{array}\right)+r\left(\begin{array}{cc}{A}_5& A_2\\ A_1& 0\\ 0& A_4\end{array}\right),\hfill \\ \hfill & r\left(\begin{array}{ccccc}{C}_5& A_5{C}_2& C_3& A_3& A_2\\ B_4& 0& 0& 0& 0\\ B_5& B_1& 0& 0& 0\\ B_2& 0& B_3& 0& 0\\ C_4& 0& 0& 0& A_4\end{array}\right)=r\left(\begin{array}{ccc}{B}_4& 0& 0\\ B_5& B_1& 0\\ B_2& 0& B_3\end{array}\right)+r\left(\begin{array}{cc}{A}_3& A_2\\ 0& A_4\end{array}\right),\hfill \end{array}$$

$$\begin{array}{cc}\hfill & r\left(\begin{array}{ccc}{C}_5& A_5{C}_2& A_2\\ B_2& 0& 0\\ B_5& B_1& 0\end{array}\right)=r\left(\begin{array}{cc}{B}_2& 0\\ B_5& B_1\end{array}\right)+r\left(\begin{array}{c}{A}_2\end{array}\right),\hfill \\ \hfill & r\left(\begin{array}{cccccccc}{C}_5& A_2& 0& 0& 0& A_5& 0& 0\\ B_2& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& -C_5& A_3& A_2& A_5& -C_3& -A_5{C}_2\\ 0& 0& B_4& 0& 0& 0& 0& 0\\ 0& 0& B_2& 0& 0& 0& B_3& 0\\ B_5& 0& B_5& 0& 0& 0& 0& B_1\\ 0& 0& -C_4& 0& A_4& 0& 0& 0\\ C_1{B}_5& 0& 0& 0& 0& A_1& 0& 0\end{array}\right)\hfill \\ \hfill & =r\left(\begin{array}{cccc}{B}_2& 0& 0& 0\\ 0& B_4& 0& 0\\ 0& B_2& B_3& 0\\ B_5& B_5& 0& B_1\end{array}\right)+r\left(\begin{array}{cccc}{A}_2& 0& 0& A_5\\ 0& A_3& A_2& A_5\\ 0& 0& A_4& 0\\ 0& 0& 0& A_1\end{array}\right).\hfill \end{array}$$

In this case, the general solution of (2) can be expressed as

$$\begin{array}{cc}\hfill & X=A_3^{†}{C}_3+U_1{B}_3+L_{A_3}{U}_2,\hfill \\ \hfill & Y=R_{A_3}{C}_3{B}_3^{†}+A_3{U}_1+U_3{R}_{B_3},\hfill \\ \hfill & Z=A_4^{†}{C}_4+W_1{B}_4+L_{A_4}{W}_2,\hfill \\ \hfill & W=R_{A_4}{C}_4{B}_4^{†}+A_4{W}_1+W_3{R}_{B_4},\hfill \\ \hfill & V=A_1^{†}{C}_1+L_{A_1}{C}_2{B}_1^{†}+L_{A_1}Q{R}_{B_1},\hfill \end{array}$$

where

$$\begin{array}{cc}\hfill & W_2={\stackrel{`}{A_1}}^{†}(\stackrel{`}{C_1}-\stackrel{`}{A_2}{W}_1\stackrel{`}{B_2}-\stackrel{`}{A_3}{U}_1\stackrel{`}{B_3}-\stackrel{`}{A_4}Q\stackrel{`}{B_4})-P_1\stackrel{`}{B_1}+L_{\stackrel{`}{A_1}}{P}_2,\hfill \\ \hfill & U_3=R_{\stackrel{`}{A_1}}(\stackrel{`}{C_1}-\stackrel{`}{A_2}{W}_1\stackrel{`}{B_2}-\stackrel{`}{A_3}{U}_1\stackrel{`}{B_3}-\stackrel{`}{A_4}Q\stackrel{`}{B_4}){\stackrel{`}{B_1}}^{†}+\stackrel{`}{A_1}{P}_1+P_3{R}_{\stackrel{`}{B_1}},\hfill \\ \hfill & W_1=A_{11}^{†}T{B}_{11}^{†}-A_{11}^{†}{A}_{22}{M}_1^{†}T{B}_{11}^{†}-A_{11}^{†}{S}_1{A}_{22}^{†}T{N}_1^{†}{B}_{22}{B}_{11}^{†}\hfill \\ \hfill & -A_{11}^{†}{S}_1{P}_4{R}_{N_1}{B}_{22}{B}_{11}^{†}+L_{A_{11}}{P}_5+P_6{R}_{B_{11}},\hfill \\ \hfill & U_1=M_1^{†}T{B}_{22}^{†}+S_1^{†}{S}_1{A}_{22}^{†}T{N}_1^{†}+L_{M_1}{L}_{S_1}{P}_7+P_8{R}_{B_{22}}+L_{M_1}{U}_4{R}_{N_1},\hfill \\ \hfill & Q=Q_{11}^0+L_{G_2}{J}_1+J_2{R}_{H_1}+L_{G_1}{J}_3{R}_{H_2},\hfill \\ \hfill & orQ=Q_{22}^0-L_{G_4}{K}_1-K_2{R}_{H_3}-L_{G_3}{K}_3{R}_{H_4},\hfill \\ \hfill & J_1=\left(\begin{array}{cc}{I}_m& 0\end{array}\right)[{\widehat{A_1}}^{†}(\overline{Q}-\widehat{C_1}{J}_3\widehat{D_1}-\widehat{C_2}{K}_3\widehat{D_2})-U_{11}\widehat{B_1}+L_{\widehat{A_1}}{U}_{12}],\hfill \\ \hfill & K_1=\left(\begin{array}{cc}0& I_m\end{array}\right)[{\widehat{A_1}}^{†}(\overline{Q}-\widehat{C_1}{J}_3\widehat{D_1}-\widehat{C_2}{K}_3\widehat{D_2})-U_{11}\widehat{B_1}+L_{\widehat{A_1}}{U}_{12}],\hfill \\ \hfill & J_2=[R_{\widehat{A_1}}(\overline{Q}-\widehat{C_1}{J}_3\widehat{D_1}-\widehat{C_2}{K}_3\widehat{D_2}){\widehat{B_1}}^{†}+\widehat{A_1}{U}_{11}+U_{21}{R}_{\widehat{B_1}}]\left(\begin{array}{c}{I}_n\\ 0\end{array}\right),\hfill \\ \hfill & K_2=[R_{\widehat{A_1}}(\overline{Q}-\widehat{C_1}{J}_3\widehat{D_1}-\widehat{C_2}{K}_3\widehat{D_2}){\widehat{B_1}}^{†}+\widehat{A_1}{U}_{11}+U_{21}{R}_{\widehat{B_1}}]\left(\begin{array}{c}0\\ I_n\end{array}\right),\hfill \\ \hfill & J_3={\widehat{A_2}}^{†}\overline{Q}{\widehat{B_2}}^{†}-{\widehat{A_2}}^{†}\widehat{A_3}{M}^{†}\overline{Q}{\widehat{B_2}}^{†}-{\widehat{A_2}}^{†}S{\widehat{A_3}}^{†}\overline{Q}{N}^{†}\widehat{B_3}{\widehat{B_2}}^{†}\hfill \\ \hfill & -{\widehat{A_2}}^{†}S{U}_{31}{R}_N\widehat{B_3}{\widehat{B_2}}^{†}+L_{\widehat{A_2}}{U}_{32}+U_{33}{R}_{\widehat{B_2}},\hfill \\ \hfill & K_3=M^{†}\overline{Q}{\widehat{B_3}}^{†}+S^{†}S{\widehat{A_3}}^{†}\overline{Q}{N}^{†}+L_M{L}_S{U}_{41}+U_{42}{R}_{\widehat{B_3}}+L_M{U}_{31}{R}_N,\hfill \end{array}$$

 and $P_i$$(i=\overline{1,8})$, $U_2$, $W_3$, $U_{11}$, $U_{12}$, $U_{21}$, $U_{31}$, $U_{32}$, $U_{33}$, $U_{41}$ and $U_{42}$ are arbitrary matrices over $\mathbb{H}$, m and n are the number of columns of $A_1$ and the number of rows of $B_1$, respectively.

Let $A_i,$ $B_i$ and $C_i$$(i=\overline{1,5})$ be given. For simplicity, consider (12) such that

$$\begin{array}{cc}\hfill & \stackrel{`}{A_1}=A_2{A}_3,\stackrel{`}{B_1}=B_4{B}_2,\stackrel{`}{A_2}=A_2,\stackrel{`}{B_2}=R_{B_3},\stackrel{`}{A_3}=L_{A_4},\stackrel{`}{B_3}=B_2,\stackrel{`}{A_4}=A_5{L}_{A_1},\hfill \\ \hfill & \stackrel{`}{B_4}=R_{B_1}{B}_5,\stackrel{`}{C_1}=C_5-[A_2{R}_{A_3}{C}_3{B}_3^{†}+A_4^{†}{C}_4{B}_2+A_5(A_1^{†}{C}_1+L_{A_1}{C}_2{B}_1^{†})B_5].\hfill \end{array}$$

(52)

Theorem 3.

The following statements are equivalent:

(1) The system (3) is solvable.

(2)

$$\begin{array}{cc}\hfill & R_{A_1}{C}_1=0,C_2{L}_{B_1}=0,A_1{C}_2=C_1{B}_1,R_{A_i}{C}_i{L}_{B_i}=0(i=3,4),\hfill \\ \hfill & R_{G_i}{L}_i=0,L_i{L}_{H_i}=0(i=\overline{1,4}),R_{\widehat{A_3}}E{L}_{\widehat{B_2}}=0.\hfill \end{array}$$

(3)

$$\begin{array}{cc}\hfill & r\left(\begin{array}{cc}{C}_1& A_1\end{array}\right)=r\left(A_1\right),r\left(\begin{array}{c}{C}_2\\ B_1\end{array}\right)=r\left(B_1\right),A_1{C}_2=C_1{B}_1,\hfill \\ \hfill & r\left(\begin{array}{cc}{C}_i& A_i\\ B_i& 0\end{array}\right)=r\left(\begin{array}{c}{A}_i\end{array}\right)+r\left(\begin{array}{c}{B}_i\end{array}\right)(i=3,4),\hfill \\ \hfill & r\left(\begin{array}{ccc}{A}_4{C}_5-C_4{B}_2& A_4{A}_2& A_4{A}_5\\ C_1{B}_5& 0& A_1\\ B_4{B}_2& 0& 0\end{array}\right)=r\left(\begin{array}{c}{B}_4{B}_2\end{array}\right)+r\left(\begin{array}{cc}{A}_4{A}_2& A_4{A}_5\\ 0& A_1\end{array}\right),\hfill \\ \hfill & r\left(\begin{array}{ccc}{C}_5{B}_3-A_2{C}_3& A_5{C}_2& A_2{A}_3\\ B_2{B}_3& 0& 0\\ B_5{B}_3& B_1& 0\end{array}\right)=r\left(\begin{array}{c}{A}_2{A}_3\end{array}\right)+r\left(\begin{array}{cc}{B}_2{B}_3& 0\\ B_5{B}_3& B_1\end{array}\right),\hfill \\ \hfill & r\left(\begin{array}{ccc}{A}_4{C}_5-C_4{B}_2& A_4{A}_2& A_4{A}_5{C}_2\\ B_4{B}_2& 0& 0\\ B_5& 0& B_1\end{array}\right)=r\left(\begin{array}{c}{A}_4{A}_2\end{array}\right)+r\left(\begin{array}{cc}{B}_5& B_1\\ B_4{B}_2& 0\end{array}\right),\hfill \\ \hfill & r\left(\begin{array}{ccc}{C}_5& A_2& A_5\\ B_2& 0& 0\\ C_1{B}_5& 0& A_1\end{array}\right)=r\left(\begin{array}{cc}{A}_2& A_5\\ 0& A_1\end{array}\right)+r\left(\begin{array}{c}{B}_2\end{array}\right),\hfill \\ \hfill & r\left(\begin{array}{ccc}{A}_4{C}_5{B}_3-A_4{A}_2{C}_3-C_4{B}_2{B}_3& A_4{A}_5& A_4{A}_2{A}_3\\ C_1{B}_5{B}_3& A_1& 0\\ B_4{B}_2{B}_3& 0& 0\end{array}\right)=r\left(\begin{array}{cc}{A}_4{A}_5& A_4{A}_2{A}_3\\ A_1& 0\end{array}\right)+r\left(\begin{array}{c}{B}_4{B}_2{B}_3\end{array}\right),\hfill \\ \hfill & r\left(\begin{array}{ccc}{C}_5{B}_3-A_2{C}_3& A_5& A_2{A}_3\\ B_2{B}_3& 0& 0\\ C_1{B}_5{B}_3& A_1& 0\end{array}\right)=r\left(\begin{array}{cc}{A}_5& A_2{A}_3\\ A_1& 0\end{array}\right)+r\left(\begin{array}{c}{B}_2{B}_3\end{array}\right),\hfill \\ \hfill & r\left(\begin{array}{ccc}{A}_4{C}_5{B}_3-A_4{A}_2{A}_3-C_4{B}_2{B}_3& A_4{A}_5{C}_2& A_4{A}_2{A}_3\\ B_5{B}_3& B_1& 0\\ B_4{B}_2{B}_3& 0& 0\end{array}\right)=r\left(\begin{array}{cc}{B}_5{B}_3& B_1\\ B_4{B}_2{B}_3& 0\end{array}\right)+r\left(\begin{array}{c}{A}_4{A}_2{A}_3\end{array}\right),\hfill \\ \hfill & r\left(\begin{array}{ccc}{C}_5& A_2& A_5{C}_2\\ B_2& 0& 0\\ B_5& 0& B_1\end{array}\right)=r\left(\begin{array}{c}{A}_2\end{array}\right)+r\left(\begin{array}{cc}{B}_2& 0\\ B_5& B_1\end{array}\right),\hfill \\ \hfill & r\left(\begin{array}{cccccc}{C}_5& A_2& 0& 0& A_5& 0\\ B_2& 0& 0& 0& 0& 0\\ 0& 0& -A_4{C}_5{B}_3+A_4{A}_2{C}_3+C_4{B}_2{B}_3& A_4{A}_2{A}_3& A_4{A}_5& -A_4{A}_5{C}_2\\ 0& 0& B_4{B}_2{B}_3& 0& 0& 0\\ B_5& 0& B_5{B}_3& 0& 0& B_1\\ C_1{B}_5& 0& 0& 0& A_1& 0\end{array}\right)\hfill \\ \hfill & =r\left(\begin{array}{ccc}{B}_2& 0& 0\\ 0& B_4{B}_2{B}_3& 0\\ B_5& B_5{B}_3& B_1\end{array}\right)+r\left(\begin{array}{ccc}{A}_2& 0& A_5\\ 0& A_4{A}_2{A}_3& A_4{A}_5\\ 0& 0& A_1\end{array}\right).\hfill \end{array}$$

In this case, the general solution of (3) can be expressed as

$$\begin{array}{cc}\hfill & X=A_3^{†}{C}_3+U_1{B}_3+L_{A_3}{U}_2,\hfill \\ \hfill & Y=R_{A_3}{C}_3{B}_3^{†}+A_3{U}_1+U_3{R}_{B_3},\hfill \\ \hfill & Z=A_4^{†}{C}_4+W_1{B}_4+L_{A_4}{W}_2,\hfill \\ \hfill & W=R_{A_4}{C}_4{B}_4^{†}+A_4{W}_1+W_3{R}_{B_4},\hfill \\ \hfill & V=A_1^{†}{C}_1+L_{A_1}{C}_2{B}_1^{†}+L_{A_1}Q{R}_{B_1},\hfill \end{array}$$

where

$$\begin{array}{cc}\hfill & U_1={\stackrel{`}{A_1}}^{†}(\stackrel{`}{C_1}-\stackrel{`}{A_2}{U}_3\stackrel{`}{B_2}-\stackrel{`}{A_3}{W}_2\stackrel{`}{B_3}-\stackrel{`}{A_4}Q\stackrel{`}{B_4})-P_1\stackrel{`}{B_1}+L_{\stackrel{`}{A_1}}{P}_2,\hfill \\ \hfill & W_1=R_{\stackrel{`}{A_1}}(\stackrel{`}{C_1}-\stackrel{`}{A_2}{U}_3\stackrel{`}{B_2}-\stackrel{`}{A_3}{W}_2\stackrel{`}{B_3}-\stackrel{`}{A_4}Q\stackrel{`}{B_4}){\stackrel{`}{B_1}}^{†}+\stackrel{`}{A_1}{P}_1+P_3{R}_{\stackrel{`}{B_1}},\hfill \\ \hfill & U_3=A_{11}^{†}T{B}_{11}^{†}-A_{11}^{†}{A}_{22}{M}_1^{†}T{B}_{11}^{†}-A_{11}^{†}{S}_1{A}_{22}^{†}T{N}_1^{†}{B}_{22}{B}_{11}^{†}\hfill \\ \hfill & -A_{11}^{†}{S}_1{P}_4{R}_{N_1}{B}_{22}{B}_{11}^{†}+L_{A_{11}}{P}_5+P_6{R}_{B_{11}},\hfill \\ \hfill & W_2=M_1^{†}T{B}_{22}^{†}+S_1^{†}{S}_1{A}_{22}^{†}T{N}_1^{†}+L_{M_1}{L}_{S_1}{P}_7+P_8{R}_{B_{22}}+L_{M_1}{U}_4{R}_{N_1},\hfill \\ \hfill & Q=Q_{11}^0+L_{G_2}{J}_1+J_2{R}_{H_1}+L_{G_1}{J}_3{R}_{H_2},\hfill \\ \hfill & orQ=Q_{22}^0-L_{G_4}{K}_1-K_2{R}_{H_3}-L_{G_3}{K}_3{R}_{H_4},\hfill \\ \hfill & J_1=\left(\begin{array}{cc}{I}_m& 0\end{array}\right)[{\widehat{A_1}}^{†}(\overline{Q}-\widehat{C_1}{J}_3\widehat{D_1}-\widehat{C_2}{K}_3\widehat{D_2})-U_{11}\widehat{B_1}+L_{\widehat{A_1}}{U}_{12}],\hfill \\ \hfill & K_1=\left(\begin{array}{cc}0& I_m\end{array}\right)[{\widehat{A_1}}^{†}(\overline{Q}-\widehat{C_1}{J}_3\widehat{D_1}-\widehat{C_2}{K}_3\widehat{D_2})-U_{11}\widehat{B_1}+L_{\widehat{A_1}}{U}_{12}],\hfill \\ \hfill & J_2=[R_{\widehat{A_1}}(\overline{Q}-\widehat{C_1}{J}_3\widehat{D_1}-\widehat{C_2}{K}_3\widehat{D_2}){\widehat{B_1}}^{†}+\widehat{A_1}{U}_{11}+U_{21}{R}_{\widehat{B_1}}]\left(\begin{array}{c}{I}_n\\ 0\end{array}\right),\hfill \\ \hfill & K_2=[R_{\widehat{A_1}}(\overline{Q}-\widehat{C_1}{J}_3\widehat{D_1}-\widehat{C_2}{K}_3\widehat{D_2}){\widehat{B_1}}^{†}+\widehat{A_1}{U}_{11}+U_{21}{R}_{\widehat{B_1}}]\left(\begin{array}{c}0\\ I_n\end{array}\right),\hfill \\ \hfill & J_3={\widehat{A_2}}^{†}\overline{Q}{\widehat{B_2}}^{†}-{\widehat{A_2}}^{†}\widehat{A_3}{M}^{†}\overline{Q}{\widehat{B_2}}^{†}-{\widehat{A_2}}^{†}S{\widehat{A_3}}^{†}\overline{Q}{N}^{†}\widehat{B_3}{\widehat{B_2}}^{†}-{\widehat{A_2}}^{†}S{U}_{31}{R}_N\widehat{B_3}{\widehat{B_2}}^{†}\hfill \\ \hfill & +L_{\widehat{A_2}}{U}_{32}+U_{33}{R}_{\widehat{B_2}},\hfill \\ \hfill & K_3=M^{†}\overline{Q}{\widehat{B_3}}^{†}+S^{†}S{\widehat{A_3}}^{†}\overline{Q}{N}^{†}+L_M{L}_S{U}_{41}+U_{42}{R}_{\widehat{B_3}}+L_M{U}_{31}{R}_N,\hfill \end{array}$$

 and $P_i$$(i=\overline{1,8})$, $U_2$, $W_3$, $U_{11}$, $U_{12}$, $U_{21}$, $U_{31}$, $U_{32}$, $U_{33}$, $U_{41}$ and $U_{42}$ are arbitrary matrices over the quaternion algebra, m and n are the number of columns of $A_1$ and the number of rows of $B_1$, respectively.

5. Conclusions

We have presented a thorough analysis of the three systems of symmetrical coupled Sylvester-like matrix Equations (1)–(3) over the quaternion algebra $\mathbb{H}$. Certain solvability conditions and the expressions of the general solutions to the systems have been derived. Some specific cases that match well-known results have been examined. Finally, an algorithm and a numerical example have been utilized to check the validity of the main findings. It is worth mentioning that the main results of this paper are valid over an arbitrary division ring.