An Exact Solution to a Quaternion Matrix Equation with an Application

Abstract

In this paper, we establish the solvability conditions and the formula of the general solution to a Sylvester-like quaternion matrix equation. As an application, we give some necessary and sufficient conditions for a system of quaternion matrix equations to be consistent, and present an expression of the general solution of the system when it is solvable. We present an algorithm and an example to illustrate the main results of this paper. The findings of this paper generalize the known results in the literature.

1. Introduction

In this paper, we mainly investigate the following matrix equation:

$$\begin{array}{cc}\hfill A_1{X}_1& B_1+A_1{X}_2{B}_2+A_2{X}_3{B}_2+A_2{X}_4{B}_3=C_c\hfill \end{array}$$

(1)

over the real quaternion algebra, $\mathbb{H}$, where $A_1,A_2,B_1,B_2,B_3$ and $C_c$ are given matrices, while $X_i(i=\overline{1,4})$ are unknown.

The quaternion algebra, $\mathbb{H}$, is a non-commutative division ring. It has many applications in computer science, orbital mechanics, signal and color image processing, and control theory, and so on (see, e.g., [1,2,3,4,5,6]).

Linear matrix equation is one of active topics in mathematics. Besides mathematics, they also have important applications in other fields, such as descriptor systems control theorem [7], neural network [8], feedback [9], and graph theory [10]. There have been a large number of papers on this topic (see, e.g., [1,2,3,4,5,11,12,13,14,15,16]). We know that the following linear matrix equation:

$$\begin{array}{c}\hfill A_1{X}_1{B}_1=C\end{array}$$

(2)

is both classical and fundamental, which was studied by many authors. For instance, Ben-Israel and Greville [17] gave a necessary and sufficient condition for the solvability to (2). Peng [18] presented some necessary and sufficient conditions for (2) to have a centrosymmetric solution by using generalized singular value decomposition. Huang [19] investigated the skew-symmetric solution and optimal approximate solution of (2). Recently, Xie and Wang [20] derived a necessary and sufficient condition for (2) to have a reducible solution. Furthermore, Xie and Wang [20] studied the following matrix equation:

$$\begin{array}{c}\hfill A_1{X}_1{B}_1+A_1{X}_2{B}_2+A_2{X}_3{B}_2=C_1\end{array}$$

(3)

which is the special case of (1). They provided some necessary and sufficient conditions for (3) to be consistent and gave an expression of its general solution when it is solvable. Motivated by the above, in this paper, we aim to establish some necessary and sufficient conditions for (1) to have a solution and derive an expression of its general solution when it is solvable. As an application of (1), we investigate the system of the following matrix equations:

$$\begin{array}{cc}\hfill & E_1{X}_1=F_1,X_1{G}_1=H_1,\hfill \\ \hfill & E_1{X}_2=F_2,X_2{G}_2=H_2,\hfill \\ \hfill & E_2{X}_3=F_3,X_3{G}_2=H_3,\hfill \\ \hfill & E_2{X}_4=F_4,X_4{G}_3=H_4,\hfill \\ \hfill E_{11}{X}_1{F}_{11}+& E_{11}{X}_2{F}_{22}+E_{22}{X}_3{F}_{22}+E_{22}{X}_4{F}_{33}=T\hfill \end{array}$$

(4)

over $\mathbb{H}$, where $X_i(i=1,\cdots ,4)$ are unknown quaternion matrices and the others are given.

The rest of this paper is structured as follows. In Section 2, we give preliminaries. In Section 3, we establish some necessary and sufficient conditions for the matrix Equation (1) to have a solution, and derive an expression of the general solution to (1) when it is solvable. as an application of (1), we derive some necessary and sufficient conditions for the system of matrix Equations (4) to have a solution as well as an expression of its general solution. Finally, we give a brief conclusion to close this paper in In Section 4.

2. Preliminaries

Throughout this paper, we denote the set of all real numbers by $\mathbb{R}$, the set of all $m\times n$ quaternion matrices by ${\mathbb{H}}^{m\times n}$, where we obtain the following:

$$\mathbb{H}=\{u_0+u_1\mathbf{i}+u_2\mathbf{j}+u_3\mathbf{k}|{\mathbf{i}}^2={\mathbf{j}}^2={\mathbf{k}}^2=\mathbf{ijk}=-1,u_0,u_1,u_2,u_3\in \mathbb{R}\}.$$

Denoted by the rank of A by $r\left(A\right)$. I and 0 represent an identity matrix and a zero matrix of appropriate sizes, respectively. An inner inverse of A is denoted by $A^{-}$ which satisfies $A{A}^{-}A=A$. $L_A$ and $R_A$ stand for the projectors $L_A=I-A^{-}A$ and $R_A=I-A{A}^{-}$, induced by A, respectively. It is easy to know that $L_A={\left(L_A\right)}^2,R_A={\left(R_A\right)}^2.$

Lemma 1

([20]). Let $A_1,A_2,B_1,B_2$ and $C_1$ be given matrices over $\mathbb{H}$ with suitable sizes. Put the following:

$$\begin{array}{c}\hfill B_3=B_1{L}_{B_2},M=R_{A_1}{A}_2,C=C_1{L}_{B_2}+R_{A_1}{C}_1,N=B_2{L}_{B_3}.\end{array}$$

Then, the following statements are equivalent:

(1)

Equation (3) is consistent.

(2)

$$\begin{array}{c}\hfill R_M{R}_{A_1}C=0,C{L}_{B_3}{L}_N=0,R_{A_1}C{L}_{B_2}=0,R_{M}C{L}_{B_3}=0.\end{array}$$

(3)

$$\begin{array}{cc}\hfill & r\left(\begin{array}{ccc}2C_1& A_2& A_1\\ B_1& 0& 0\\ B_2& 0& 0\end{array}\right)=r(A_2,A_1)+r\left(\begin{array}{c}{B}_1\\ B_2\end{array}\right),r\left(\begin{array}{ccc}2C_1& A_2& A_1\\ B_2& 0& 0\end{array}\right)=r(A_2,A_1)+r\left(\begin{array}{c}{B}_2\end{array}\right),\hfill \\ \hfill & r\left(\begin{array}{cc}2C_1& A_1\\ B_1& 0\\ B_2& 0\end{array}\right)=r\left(A_1\right)+r\left(\begin{array}{c}{B}_1\\ B_2\end{array}\right),r\left(\begin{array}{cc}{C}_1& A_1\\ B_2& 0\end{array}\right)=r\left(A_1\right)+r\left(\begin{array}{c}{B}_2\end{array}\right).\hfill \end{array}$$

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

$$\begin{array}{cc}\hfill & X_1=A_1^{-}C{B}_3^{-}+L_{A_1}{V}_1+V_2{R}_{B_3},\hfill \\ \hfill & X_2=A_1^{-}(C_1-A_1{X}_1{B}_1-A_2{X}_3{B}_2)B_2^{-}+T_1{R}_{B_2}-L_{A_1}{T}_2,\hfill \\ \hfill & X_3=M^{-}C{B}_2^{-}+L_M{U}_1+U_2{R}_{B_2},\hfill \end{array}$$

where $U_1,U_2,V_1,V_2,T_1$ and $T_2$ are arbitrary matrices over $\mathbb{H}$, with appropriate sizes.

The following lemma is due to Marsaglia and Styan [21], which can be generalized to $\mathbb{H}$.

Lemma 2

[21]). Let $A\in {\mathbb{H}}^{m\times n},$$B\in {\mathbb{H}}^{m\times k},$$C\in {\mathbb{H}}^{l\times n}$, $D\in {\mathbb{H}}^{j\times k}$ and $E\in {\mathbb{H}}^{l\times i}$ be given. Then, we have the following rank equality:

$$r\left(\begin{array}{cc}A& B{L}_D\\ R_{E}C& 0\end{array}\right)=r\left(\begin{array}{ccc}A& B& 0\\ C& 0& E\\ 0& D& 0\end{array}\right)-r\left(D\right)-r\left(E\right).$$

Lemma 3

([22]). Let $A_{ii},B_{ii}$ and $C_{ii}(i=\overline{1,2})$ be given matrices over $\mathbb{H}$ with appropriate sizes. Put the following:

$$\begin{array}{c}{A}_1=A_{22}{L}_{A_{11}},B_1=R_{B_{11}}{B}_{22},C_1=C_{22}-A_{22}{A}_{11}^{-}{C}_{11}{B}_{11}^{-}{B}_{22},D_1=R_{A_1}{A}_{22},\\ \varphi =A_{11}^{-}{C}_{11}{B}_{11}^{-}+L_{A_{11}}{A}_1^{-}{C}_1{B}_{22}^{-}-L_{A_{11}}{A}_1^{-}{A}_{22}{D}_1^{-}{R}_{A_1}{C}_1{B}_{22}^{-}+D_1^{-}{R}_{A_1}{C}_1{B}_1^{-}{R}_{B_{11}}.\end{array}$$

Then, the system of matrix equations $A_{ii}X{B}_{ii}=C_{ii}(i=\overline{1,2})$ has a solution if—and only if—the following is true:

$$\begin{array}{cc}& R_{A_{ii}}{C}_{ii}=0,C_{ii}{L}_{B_{ii}}=0,(i=\overline{1,2}),R_{A_1}{C}_1{L}_{B_1}=0.\hfill \end{array}$$

In this case, the general solution to the system can be expressed as follows:

$$X=\varphi +L_{A_{11}}{L}_{A_1}{U}_1+U_2{R}_{B_1}{R}_{B_{11}}+L_{A_{11}}{U}_3{R}_{B_{22}}+L_{A_{22}}{U}_4{R}_{B_{11}},$$

where $U_i(i=\overline{1,4})$ are arbitrary matrices over $\mathbb{H}$, with appropriate sizes.

Lemma 4

([23]). Let $A_1\in {\mathbb{H}}^{m_1\times n_1},B_1\in {\mathbb{H}}^{r_1\times s_1},C_1\in {\mathbb{H}}^{m_1\times r_1}$ and $C_2\in {\mathbb{H}}^{n_1\times s_1}$ be given. Then, we obtain the following system:

$$\begin{array}{c}\hfill A_1{X}_1=C_1,X_1{B}_1=C_2\end{array}$$

(5)

which is consistent if—and only if—the following is true:

$$R_{A_1}{C}_1=0,C_2{L}_{B_1}=0,A_1{C}_2=C_1{B}_1.$$

Under these conditions, a general solution to (5) can be expressed as follows:

$$X_1=A_1^{-}{C}_1+L_{A_1}{C}_2{B}_1^{-}+L_{A_1}{U}_1{R}_{B_1},$$

where $U_1$ is an any matrix with conformable dimension.

Lemma 5

([24]). Consider the following matrix equation:

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

(6)

over $\mathbb{H}$, where $A_1,B_1,C_3,D_3,C_4,D_4$ and $E_1$ be given matrices of suitable sizes. Put the following:

$$\begin{array}{cc}\hfill & A=R_{A_1}{C}_3,B=D_3{L}_{B_1},C=R_{A_1}{C}_4,D=D_4{L}_{B_1},\hfill \\ \hfill & E=R_{A_1}{E}_1{L}_{B_1},M=R_{A}C,N=D{L}_B,S=C{L}_M.\hfill \end{array}$$

Then, the following statements are equivalent:

(1)

The matrix Equation (6) has a solution.

(2)

$$\begin{array}{c}\hfill R_M{R}_{A}E=0,E{L}_B{L}_N=0,R_{A}E{L}_D=0,R_{C}E{L}_B=0.\end{array}$$

(3)

$$\begin{array}{cc}\hfill & r\left(\begin{array}{cccc}{E}_1& C_4& C_3& A_1\\ B_1& 0& 0& 0\end{array}\right)=r\left(B_1\right)+r(C_4,C_3,A_1),\hfill \\ \hfill & r\left(\begin{array}{cc}{E}_1& A_1\\ D_3& 0\\ D_4& 0\\ B_1& 0\end{array}\right)=r\left(\begin{array}{c}{D}_3\\ D_4\\ B_1\end{array}\right)+r\left(A_1\right),\hfill \\ \hfill & r\left(\begin{array}{ccc}{E}_1& C_3& A_1\\ D_4& 0& 0\\ B_1& 0& 0\end{array}\right)=r(C_3,A_1)+r\left(\begin{array}{c}{D}_4\\ B_1\end{array}\right),\hfill \\ \hfill & r\left(\begin{array}{ccc}{E}_1& C_4& A_1\\ D_3& 0& 0\\ B_1& 0& 0\end{array}\right)=r(C_4,A_1)+r\left(\begin{array}{c}{D}_3\\ B_1\end{array}\right).\hfill \end{array}$$

In this case, the general solution to the matrix Equation (6) can be expressed as follows:

$$\begin{array}{cc}\hfill & X_1=A_1^{-}(E_1-C_3{X}_3{D}_3-C_4{X}_4{D}_4)-A_1^{-}{T}_7{B}_1+L_{A_1}{T}_6,\hfill \\ \hfill & X_2=R_{A_1}(E_1-C_3{X}_3{D}_3-C_4{X}_4{D}_4)B_1^{-}+A_1{A}_1^{-}{T}_7+T_8{R}_{B_1},\hfill \\ \hfill & X_3=A^{-}E{B}^{-}-A^{-}C{M}^{-}E{B}^{-}-A^{-}S{C}^{-}E{N}^{-}D{B}^{-}-A^{-}S{T}_2{R}_{N}D{B}^{-}+L_A{T}_4+T_5{R}_B,\hfill \\ \hfill & X_4=M^{-}E{D}^{-}+S^{-}S{C}^{-}E{N}^{-}+L_M{L}_S{T}_1+L_M{T}_2{R}_N+T_3{R}_D,\hfill \end{array}$$

where $T_1,\dots ,T_8$ are arbitrary matrices of appropriate sizes over $\mathbb{H}$.

Lemma 6

([22]). Let $A_1,B_1$ and $C_1$ be given matrices over $\mathbb{H}$ with suitable sizes. Then, the matrix equation $A_1{X}_1{B}_1=C_1$ is consistent if—and only if—$R_{A_1}{C}_1=0,C_1{L}_{B_1}=0$. In this case, the general solution of the matrix equation can be expressed as follows:

$$X_1=A_1^{-}{C}_1{B}_1^{-}+L_{A_1}V+U{R}_{B_1},$$

where U and V are any matrices with compatible dimensions over $\mathbb{H}$.

3. The General Solution to the Matrix Equation (1)

For convenience, we define the notation as follows: let $A_i,B_j(i=\overline{1,2},j=\overline{1,3})$ be given matrices of suitable sizes over $\mathbb{H}$ and put the following:

$$\begin{array}{cc}\hfill & B_4=B_1{L}_{B_2},B_5=B_2{L}_{B_4},A_{33}=R_{A_1}{A}_2,C_{44}=C_c{L}_{B_2}+R_{A_1}{C}_c,B_{33}=B_3{L}_{B_2},C_{33}=R_{A_1}{C}_c{L}_{B_2},\hfill \\ & A_{11}=A_2+A_{33},B_{11}=B_3{L}_{B_4}{L}_{B_5},C_{11}=(I+R_{A_1})C_c{L}_{B_4}{L}_{B_5},A_{22}=R_{A_{33}}{A}_2,B_{22}=B_3{L}_{B_2}{L}_{B_4},\hfill \\ \hfill & C_{22}=R_{A_{33}}{C}_{44}{L}_{B_4},M_1=A_{22}{L}_{A_{11}},N_1=R_{B_{11}}{B}_{22},C_1=C_{22}-A_{22}{A}_{11}^{-}{C}_{11}{B}_{11}^{-}{B}_{22},D_1=R_{A_1}{A}_{22},\hfill \\ & \varphi =A_{11}^{-}{C}_{11}{B}_{11}^{-}+L_{A_{11}}{M}_1^{-}{C}_1{B}_{22}^{-}-L_{A_{11}}{M}_1^{-}{A}_{22}{D}_1^{-}{R}_{M_1}{C}_1{B}_{22}^{-}+D_1^{-}{R}_{M_1}{C}_1{N}_1^{-}{R}_{B_{11}},\hfill \\ \hfill & M_2=\left[L_{A_{11}}{L}_{M_1},L_{A_{33}}\right],N_2=\left[\begin{array}{c}{R}_{N_1}{R}_{B_{11}}\\ R_{B_{33}}\end{array}\right],A=R_{M_2}{L}_{A_{11}},B=R_{B_{22}}{L}_{N_2},C=R_{M_2}{L}_{A_{22}},\hfill \\ \hfill & D=R_{B_{11}}{L}_{N_2},E_1=A_{33}^{-}{C}_{33}{B}_{33}^{-}-\varphi ,E=R_{M_2}{E}_1{L}_{N_2},M=R_{A}C,N=D{L}_B,S=C{L}_M,\hfill \end{array}$$

(7)

and

$$S_1=\left[I_m,0\right],S_2=\left[\begin{array}{c}{I}_n\\ 0\end{array}\right],S_3=\left[0,I_m\right],S_4=\left[\begin{array}{c}0\\ I_n\end{array}\right],$$

where $I_m$ and $I_n$ denote the unit matrices of order n and m, respectively. Then, we obtain the main theorem of this paper.

Theorem 1.

Consider (1) with the notation in (7). The following statements are equivalent:

(1)

The matrix Equation (1) has a solution.

(2)

$$\begin{array}{cc}\hfill & \begin{array}{c}\hfill R_{A_{33}}{R}_{A_1}{C}_{44}=0,R_{A_{11}}{C}_{11}=0,C_{ii}{L}_{B_{ii}}=0,(i=\overline{1,3}),\end{array}\hfill \end{array}$$

(8)

$$\begin{array}{cc}\hfill & \begin{array}{cc}\hfill & R_{M_1}{C}_1{L}_{N_1}=0,R_{A}E{L}_D=0,R_M{R}_{A}E=0,E{L}_B{L}_N=0,R_{C}E{L}_B=0.\hfill \end{array}\hfill \end{array}$$

(9)

(3)

$$\begin{array}{cc}\hfill & \begin{array}{cc}\hfill & r\left(\begin{array}{ccc}2C_c& A_2& A_1\\ B_2& 0& 0\end{array}\right)=r(A_2,A_1)+r\left(B_2\right),\hfill \end{array}\hfill \end{array}$$

(10)

$$\begin{array}{cc}\hfill & \begin{array}{cc}\hfill & r\left(\begin{array}{ccc}2C_c& A_2& A_1\\ B_1& 0& 0\\ B_2& 0& 0\end{array}\right)=r(A_2,A_1)+r\left(\begin{array}{c}{B}_1\\ B_2\end{array}\right),\hfill \end{array}\hfill \end{array}$$

(11)

$$\begin{array}{cc}\hfill & \begin{array}{cc}\hfill & r\left(\begin{array}{cc}2C_c& A_1\\ B_3& 0\\ B_2& 0\\ B_1& 0\end{array}\right)=r\left(A_1\right)+r\left(\begin{array}{c}{B}_3\\ B_2\\ B_1\end{array}\right),\hfill \end{array}\hfill \end{array}$$

(12)

$$\begin{array}{cc}\hfill & \begin{array}{cc}\hfill & r\left(\begin{array}{ccc}2C_c& A_2& A_1\\ B_3& 0& 0\\ B_2& 0& 0\\ B_1& 0& 0\end{array}\right)=r(A_2,A_1)+r\left(\begin{array}{c}{B}_3\\ B_2\\ B_1\end{array}\right),\hfill \end{array}\hfill \end{array}$$

(13)

$$\begin{array}{cc}\hfill & \begin{array}{cc}\hfill & r\left(\begin{array}{cc}{C}_c& A_1\\ B_3& 0\\ B_2& 0\end{array}\right)=r\left(A_1\right)+r\left(\begin{array}{c}{B}_3\\ B_2\end{array}\right),\hfill \end{array}\hfill \end{array}$$

(14)

$$\begin{array}{cc}\hfill & \begin{array}{cc}\hfill & r\left(\begin{array}{cccccc}2C_c& 0& 0& A_2& A_1& 0\\ B_3& 0& B_3& 0& 0& 0\\ 0& A_2& -2C_c& 0& 0& A_1\\ B_1& 0& 0& 0& 0& 0\\ 0& 0& B_1& 0& 0& 0\\ 0& 0& B_2& 0& 0& 0\\ B_2& 0& 0& 0& 0& 0\end{array}\right)=r\left(\begin{array}{cccc}{A}_2& A_1& 0& 0\\ 0& 0& A_2& A_1\end{array}\right)+r\left(\begin{array}{cc}{B}_3& B_3\\ B_2& 0\\ B_1& 0\\ 0& B_2\\ 0& B_1\end{array}\right),\hfill \end{array}\hfill \end{array}$$

(15)

$$\begin{array}{cc}\hfill & \begin{array}{cc}\hfill & r\left(\begin{array}{ccccc}0& B_3& B_3& 0& 0\\ -2A_2& 2C_c& 0& A_1& 0\\ A_2& 0& C_c& 0& A_1\\ 0& B_2& 0& 0& 0\\ 0& B_1& 0& 0& 0\\ 0& 0& B_2& 0& 0\end{array}\right)=r\left(\begin{array}{ccc}0& A_1& 0\\ A_2& 0& A_1\end{array}\right)+r\left(\begin{array}{cc}{B}_3& B_3\\ B_2& 0\\ B_1& 0\\ 0& B_2\end{array}\right),\hfill \end{array}\hfill \end{array}$$

(16)

$$\begin{array}{c}\hfill r\left(\begin{array}{ccccccccc}0& 0& -B_3& B_3& B_3& 0& 0& 0& 0\\ 0& 0& 2C_c& 0& 0& A_2& 0& A_1& 0\\ A_2& 0& 0& 2C_c& 0& 0& 0& 0& A_1\\ 0& A_2& 0& 0& C_c& 0& A_1& 0& 0\\ 0& 0& 0& 0& B_2& 0& 0& 0& 0\\ 0& 0& B_1& 0& 0& 0& 0& 0& 0\\ 0& 0& B_2& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& B_2& 0& 0& 0& 0& 0\\ 0& 0& 0& B_1& 0& 0& 0& 0& 0\end{array}\right)\end{array}$$

$$\begin{array}{cc}\hfill & \begin{array}{c}\hfill =r\left(\begin{array}{cccccc}{A}_2& A_1& 0& 0& 0& 0\\ 0& 0& A_2& A_1& 0& 0\\ 0& 0& 0& 0& A_2& A_1\end{array}\right)+r\left(\begin{array}{ccc}{B}_3& B_3& B_3\\ B_2& 0& 0\\ B_1& 0& 0\\ 0& B_2& 0\\ 0& B_1& 0\\ 0& 0& B_2\end{array}\right),\end{array}\hfill \end{array}$$

(17)

$$\begin{array}{cc}\hfill & \begin{array}{cc}\hfill & r\left(\begin{array}{cccccccc}0& B_3& B_3& 0& 0& 0& 0& 0\\ 0& 0& -B_3& B_3& 0& 0& 0& 0\\ 2A_2& 0& 2C_c& 0& 0& 0& A_1& 0\\ 0& 2C_c& 0& 0& 0& A_2& 0& A_1\\ A_2& 0& 0& C_c& A_1& 0& 0& 0\\ 0& B_2& 0& 0& 0& 0& 0& 0\\ 0& B_1& 0& 0& 0& 0& 0& 0\\ 0& 0& B_2& 0& 0& 0& 0& 0\\ 0& 0& B_1& 0& 0& 0& 0& 0\\ 0& 0& 0& B_2& 0& 0& 0& 0\end{array}\right)\hfill \\ \hfill & =r\left(\begin{array}{ccccc}0& A_2& 0& A_1& 0\\ 2A_2& 0& 0& 0& A_1\\ A_2& 0& A_1& 0& 0\end{array}\right)+r\left(\begin{array}{ccc}{B}_3& B_3& 0\\ 0& B_3& B_3\\ B_2& 0& 0\\ B_1& 0& 0\\ 0& B_2& 0\\ 0& B_1& 0\\ 0& 0& B_2\end{array}\right),\hfill \end{array}\hfill \end{array}$$

(18)

$$\begin{array}{cc}\hfill & \begin{array}{cc}\hfill & r\left(\begin{array}{ccccccccc}0& 0& B_3& B_3& B_3& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& B_3& 0& 0& 0\\ 0& 0& 2C_c& 0& 0& 0& A_1& A_2& 0\\ A_2& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& A_1\\ 0& A_2& 0& 0& 0& C_c& 0& 0& 0\\ 0& 0& B_1& 0& 0& 0& 0& 0& 0\\ 0& 0& B_2& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& B_1& 0& 0& 0& 0& 0\\ 0& 0& 0& B_2& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& B_1& 0& 0& 0& 0\\ 0& 0& 0& 0& B_2& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& B_2& 0& 0& 0\end{array}\right)\hfill \end{array}\hfill \end{array}$$

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill & =r\left(\begin{array}{ccccc}0& 0& A_2& A_1& 0\\ A_2& 0& 0& 0& 0\\ 0& 0& 0& 0& A_1\\ 0& A_2& 0& 0& 0\end{array}\right)+r\left(\begin{array}{cccc}{B}_3& B_3& B_3& 0\\ 0& 0& 0& B_3\\ B_2& 0& 0& 0\\ B_1& 0& 0& 0\\ 0& B_2& 0& 0\\ 0& B_1& 0& 0\\ 0& 0& B_1& 0\\ 0& 0& B_2& 0\\ 0& 0& 0& B_2\end{array}\right).\hfill \end{array}\end{array}$$

(19)

In this case, the general solution to (1) can be expressed as follows:

$$\begin{array}{cc}\hfill & X_1=A_1^{-}\left[(C_c-A_2{X}_4{B}_3)L_{B_2}+R_{A_1}(C_c-A_2{X}_4{B}_3)\right]B_4^{-}+L_{A_1}{V}_1+V_2{R}_{B_4},\hfill \\ \hfill & X_2=A_1^{-}(C_c-A_1{X}_1{B}_1-A_2{X}_3{B}_2-A_2{X}_4{B}_3)B_2^{-}+V_3{R}_{B_2}-L_{A_1}{V}_4,\hfill \\ \hfill & X_3=M^{-}\left[(C_c-A_2{X}_4{B}_3)L_{B_2}+R_{A_1}(C_c-A_2{X}_4{B}_3)\right]B_2^{-}+L_M{V}_5+V_6{R}_{B_2},\hfill \\ \hfill & X_4=\varphi +L_{A_{11}}{L}_{M_1}{U}_1+U_2{R}_{N_1}{R}_{B_{11}}+L_{A_{11}}{U}_3{R}_{B_{22}}+L_{A_{22}}{U}_4{R}_{B_{11}},\hfill \\ \hfill or\\ \hfill & X_4=A_{33}^{-}{C}_{33}{B}_{33}^{-}+L_{A_{33}}{U}_5+U_6{R}_{B_{33}},\hfill \end{array}$$

(20)

where

$$\begin{array}{cc}\hfill & U_1=S_1\left[M_2^{-}\left(E_1-L_{A_{11}}{U}_3{R}_{B_{22}}-L_{A_{22}}{U}_4{R}_{B_{11}}\right)-M_2^{-}{T}_7{N}_2+L_{M_2}{T}_6\right],\hfill \\ \hfill & U_2=\left[R_{M_2}\left(E_1-L_{A_{11}}{U}_3{R}_{B_{22}}-L_{A_2}{U}_4{R}_{B_{11}}\right)N_2^{-}+M_2{M}_2^{-}{T}_7+T_8{R}_{N_2}\right]S_2,\hfill \\ \hfill & U_3=A^{-}E{B}^{-}-A^{-}C{M}^{-}{R}_{A}E{B}^{-}-A^{-}S{C}^{-}E{L}_B{N}^{-}D{B}^{-}-A^{-}S{T}_3{R}_{N}D{B}^{-}\hfill \\ \hfill & +L_A{T}_1+T_2{R}_B,\hfill \\ \hfill & U_4=M^{-}{R}_{A}E{D}^{-}+L_M{S}^{-}S{C}^{-}E{L}_B{N}^{-}+L_M{L}_S{T}_4+L_M{T}_3{R}_N+T_5{R}_D,\hfill \\ \hfill & U_5=-S_3\left[M_2^{-}\left(E_1-L_{A_{11}}{U}_3{R}_{B_{22}}-L_{A_{22}}{U}_4{R}_{B_{11}}\right)-M_2^{-}{T}_7{N}_2+L_{M_2}{T}_6\right],\hfill \\ \hfill & U_6=-\left[R_{M_2}\left(E_1-L_{A_{11}}{U}_3{R}_{B_{22}}-L_{A_{22}}{U}_4{R}_{B_{11}}\right)N_2^{-}+M_2{M}_2^{-}{T}_7+T_8{R}_{N_2}\right]S_4,\hfill \end{array}$$

and $V_i(i=\overline{1,6}),$$T_i(i=\overline{1,8})$ are arbitrary matrices over $\mathbb{H}$, with appropriate sizes.

Proof. 

$\left(1\right)\iff \left(2\right)$: It is easy to know that Equation (1) can be written as follows:

$$\begin{array}{c}\hfill A_1{X}_1{B}_1+A_1{X}_2{B}_2+A_2{X}_3{B}_2=C_c-A_2{X}_4{B}_3.\end{array}$$

(21)

Clearly, Equation (1) is solvable if—and only if—Equation (21) is consistent. By Lemma 1, we obtain that (21) is solvable if—and only if—there exists $X_4$ in (21) such that we obtain the following:

$$\begin{array}{cc}\hfill & R_{A_{33}}{R}_{A_1}\left[(C_c-A_2{X}_4{B}_3)L_{B_2}+R_{A_1}(C_c-A_2{X}_4{B}_3)\right]=0,\hfill \\ \hfill & \left[(C_c-A_2{X}_4{B}_3)L_{B_2}+R_{A_1}(C_c-A_2{X}_4{B}_3)\right]L_{B_3}{L}_{B_5}=0,\hfill \\ & R_{A_{33}}\left[(C_c-A_2{X}_4{B}_3)L_{B_2}+R_{A_1}(C_c-A_2{X}_4{B}_3)\right]L_{B_4}=0,\hfill \\ \hfill & R_{A_1}\left[(C_c-A_2{X}_4{B}_3)L_{B_2}+R_{A_1}(C_c-A_2{X}_4{B}_3)\right]L_{B_2}=0,\hfill \end{array}$$

i.e.,

$$\begin{array}{cc}\hfill & R_{A_{33}}{R}_{A_1}{C}_{44}=0,\hfill \end{array}$$

(22)

$$\begin{array}{c}\hfill A_{ii}{X}_4{B}_{ii}=C_{ii}(i=1,2),\end{array}$$

(23)

and

$$\begin{array}{c}\hfill A_{33}{X}_4{B}_{33}=C_{33},\end{array}$$

(24)

respectively. Moreover, when (22) is solvable, we obtain the following:

$$\begin{array}{cc}\hfill & X_1=A_1^{-}\left[(C_c-A_2{X}_4{B}_3)L_{B_2}+R_{A_1}(C_c-A_2{X}_4{B}_3)\right]B_4^{-}+L_{A_1}{V}_1+V_2{R}_{B_4},\hfill \\ \hfill & X_2=A_1^{-}(C_c-A_1{X}_1{B}_1-A_2{X}_3{B}_2-A_2{X}_4{B}_3)B_2^{-}+V_3{R}_{B_2}-L_{A_1}{V}_4,\hfill \\ \hfill & X_3=M^{-}\left[(C_c-A_2{X}_4{B}_3)L_{B_2}+R_{A_1}(C_c-A_2{X}_4{B}_3)\right]B_2^{-}+L_M{V}_5+V_6{R}_{B_2},\hfill \end{array}$$

where $B_4,B_5$, $A_{33}$ and M are defined by (7), $V_i(i=\overline{1,6})$, which are arbitrary matrices over $\mathbb{H}$, with appropriate sizes.

Hence, the matrix Equation (21) is solvable if—and only if—(22) holds, and there exists $X_4$, such that both (23) and (24) are solvable.

Next, we consider the common solution of (23) and (24). On the ond hand, by Lemma 3, the system (23) is solvable if—and only if—the following is true:

$$\begin{array}{cc}\hfill & R_{A_{ii}}{C}_{ii}=0,C_{ii}{L}_{B_{ii}}=0(i=1,2),R_{M_1}{C}_1{L}_{N_1}=0,\hfill \end{array}$$

(25)

in which case, the general solution of (23) can be expressed as follows:

$$\begin{array}{c}\hfill X_4=\varphi +L_{A_{11}}{L}_{M_1}{U}_1+U_2{R}_{N_1}{R}_{B_{11}}+L_{A_{11}}{U}_3{R}_{B_{22}}+L_{A_{22}}{U}_4{R}_{B_{11}},\end{array}$$

(26)

where $A_{ii}(i=1,2)$, $M_1,N_1$ and $C_1$ are given by (7), and $U_i(i=\overline{1,4})$ are arbitrary matrices over $\mathbb{H}$, with appropriate sizes. On the other hand, in view of Lemma 6, (24) is solvable if—and only if—the following is true:

$$\begin{array}{c}\hfill R_{A_{33}}{C}_{33}=0,C_{33}{L}_{B_{33}}=0,\end{array}$$

(27)

in which case, the general solution of (24) can be expressed as follows:

$$\begin{array}{c}\hfill X_4=A_{33}^{-}{C}_{33}{B}_{33}^{-}+L_{A_{33}}{U}_5+U_6{R}_{B_{33}},\end{array}$$

(28)

where $A_{33},B_{33}$ and $C_{33}$ are given by (7), and $U_5$ and $U_6$ are arbitrary matrices over $\mathbb{H}$, with appropriate sizes.

Clearly, the matrix Equations (23) and (24) have a common solution if—and only if—$X_4$ of (26) is equal to $X_4$ of (28). Letting $X_4$ of (26) be the one of (28), yielding the following:

$$(L_{A_{11}}{L}_{M_1},L_{A_{33}})\left(\begin{array}{c}{U}_1\\ -U_5\end{array}\right)+(U_2,-U_6)\left(\begin{array}{c}{R}_{N1}{R}_{B_{11}}\\ R_{B_{33}}\end{array}\right)+L_{A_{11}}{U}_3{R}_{B_{22}}+L_{A_{22}}{U}_4{R}_{B_{11}}=E_1,$$

i.e.,

$$\begin{array}{c}\hfill M_2\left(\begin{array}{c}{U}_1\\ -U_5\end{array}\right)+(U_2,-U_6)N_2+L_{A_{11}}{U}_3{R}_{B_{22}}+L_{A_{22}}{U}_4{R}_{B_{11}}=E_1.\end{array}$$

(29)

It follows from Lemma 5 that Equation (29) has a solution if—and only if—the following is true:

$$\begin{array}{c}\hfill R_{A}E{L}_D=0,R_M{R}_{A}E=0,E{L}_B{L}_N=0,R_{C}E{L}_B=0.\end{array}$$

(30)

In this case, the general solution to (29) can be expressed as follows:

$$\begin{array}{cc}\hfill & U_1=S_1\left[M_2^{-}\left(E_1-L_{A_{11}}{U}_3{R}_{B_{22}}-L_{A_{22}}{U}_4{R}_{B_{11}}\right)-M_2^{-}{T}_7{N}_2+L_{M_2}{T}_6\right],\hfill \\ \hfill & U_2=\left[R_{M_2}\left(E_1-L_{A_{11}}{U}_3{R}_{B_{22}}-L_{A_2}{U}_4{R}_{B_{11}}\right)N_2^{-}+M_2{M}_2^{-}{T}_7+T_8{R}_{N_2}\right]S_2,\hfill \\ \hfill & U_3=A^{-}E{B}^{-}-A^{-}C{M}^{-}{R}_{A}E{B}^{-}-A^{-}S{C}^{-}E{L}_B{N}^{-}D{B}^{-}-A^{-}S{T}_3{R}_{N}D{B}^{-}\hfill \\ \hfill & +L_A{T}_1+T_2{R}_B,\hfill \\ \hfill & U_4=M^{-}{R}_{A}E{D}^{-}+L_M{S}^{-}S{C}^{-}E{L}_B{N}^{-}+L_M{L}_S{T}_4+L_M{T}_3{R}_N+T_5{R}_D,\hfill \\ \hfill & U_5=-S_3\left[M_2^{-}\left(E_1-L_{A_{11}}{U}_3{R}_{B_{22}}-L_{A_{22}}{U}_4{R}_{B_{11}}\right)-M_2^{-}{T}_7{N}_2+L_{M_2}{T}_6\right],\hfill \\ \hfill & U_6=-\left[R_{M_2}\left(E_1-L_{A_{11}}{U}_3{R}_{B_{22}}-L_{A_{22}}{U}_4{R}_{B_{11}}\right)N_2^{-}+M_2{M}_2^{-}{T}_7+T_8{R}_{N_2}\right]S_4,\hfill \end{array}$$

where $M_1,M_2,N_1,N_2,A,B,D,E,S,M,N$ are defined as (7), and $T_j(j=\overline{1,8})$ are arbitrary matrices over $\mathbb{H}$, with appropriate sizes.

So far, we have shown that Equation (21) is solvable if—and only if—(22), (25), (27), and (30) hold. In this case, the general solution of (21) can be expressed as (20).

We now show that $R_{A_{33}}{C}_{33}=0\iff R_{A_{33}}{R}_{A_1}{C}_{44}=0$ and $R_{A_{22}}{C}_{22}=0\iff R_{A_{11}}{C}_{11}=0.$

In fact, it follows from Lemma 2 that we obtain the following:

$$\begin{array}{cc}\hfill & R_{A_{33}}{C}_{33}=0\iff r\left(R_{A_{33}}{C}_{33}\right)=0\iff r(C_{33},A_{33})=r\left(A_{33}\right)\hfill \\ \hfill & \iff r(R_{{}_{A_1}}{C}_c{L}_{B_2},R_{A_1}{A}_2)=r\left(R_{A_1}{A}_2\right)\iff r\left(\begin{array}{ccc}{C}_c& A_2& A_1\\ B_2& 0& 0\end{array}\right)=r(A_2,A_1)+r\left(B_2\right)\hfill \end{array}$$

(31)

and

$$\begin{array}{cc}\hfill & R_{A_{33}}{R}_{A_1}{C}_{44}=0\iff r\left(R_{A_{33}}{R}_{A_1}(C_c{L}_{B_2}+R_{A_1}{C}_c)\right)=0\hfill \\ \hfill & \iff r(\left(C_c{L}_{B_2}+R_{A_1}{C}_c\right),A_2,A_1)=r(A_2,A_1)\hfill \\ \hfill & \iff r\left(\begin{array}{ccc}2C_c& A_2& A_1\\ B_2& 0& 0\end{array}\right)=r(A_2,A_1)+r\left(B_2\right)\hfill \end{array}$$

(32)

yielding $R_{A_{33}}{C}_{33}=0\iff R_{A_{33}}{R}_{A_1}{C}_{44}=0.$

On the other hand, we have the following:

$$\begin{array}{cc}\hfill & R_{A_{11}}{C}_{11}=0\iff r\left(R_{A_{11}}{C}_{11}\right)=0\iff r(C_{11},A_{11})=r\left(A_{11}\right)\hfill \\ \hfill & \iff r[(I+R_{A_1})C_c{L}_{B_4}{L}_{B_5},(I+R_{A_1})A_2]=r\left[(I+R_{A_1})A_2\right]\hfill \\ \hfill & \iff r\left(\begin{array}{ccc}I& 0& 0\\ 0& (I+R_{A_1})C_c{L}_{B_4}{L}_{B_5}& (I+R_{A_1})A_2\end{array}\right)=r\left(\begin{array}{cc}I& 0\\ 0& (I+R_{A_1})A_2\end{array}\right)\hfill \\ \hfill & \iff r\left(\begin{array}{cccc}I& -C_c& -A_2& 0\\ I& C_c& A_2& A_1\\ 0& B_2& 0& 0\\ 0& B_1& 0& 0\end{array}\right)=r\left(\begin{array}{ccc}I& -A_2& 0\\ I& A_2& A_1\end{array}\right)+r\left(\begin{array}{c}{B}_2\\ B_1\end{array}\right)\hfill \\ \hfill & \iff r\left(\begin{array}{ccc}2C_c& A_2& A_1\\ B_2& 0& 0\\ B_1& 0& 0\end{array}\right)=r(A_2,A_1)+r\left(\begin{array}{c}{B}_2\\ B_1\end{array}\right)\iff \left(11\right).\hfill \end{array}$$

(33)

Similarly, we can show that $R_{A_{22}}{C}_{22}=0\iff$ (11). Hence, $R_{A_{11}}{C}_{11}=0\iff R_{A_{22}}{C}_{22}=0$.

So far, we have proved that (22), (25), (27), and (30) hold if—and only if—(8) and (9) hold.

To sum up, Equation (21), and thus Equation (1), are consistent if—and only if—(8) and (9) hold.

$\left(2\right)\iff \left(3\right)$: We first show that (8) holds if—and only if—(10)–(14) hold. It follows from (32) and (33) that (22) $\iff$ (10) and $R_{A_{11}}{C}_{11}=0\iff$ (11). Next, we prove that $C_{11}{L}_{B_{11}}=0\iff$ (12). By Lemma 2, as follows:

$$\begin{array}{cc}\hfill & C_{11}{L}_{B_{11}}=0\iff r\left(C_{11}{L}_{B_{11}}\right)=0\iff r\left(\begin{array}{c}{C}_{11}\\ B_{11}\end{array}\right)=r\left(B_{11}\right)\hfill \\ \hfill & \iff r\left(\begin{array}{c}(I+R_{A_1})C_c\\ B_3\\ B_2\\ B_1\end{array}\right)=r\left(\begin{array}{c}{B}_3\\ B_2\\ B_1\end{array}\right)\iff r\left(\begin{array}{cc}I& 0\\ I& (I+R_{A_1})C_c\\ 0& B_3\\ 0& B_2\\ 0& B_1\end{array}\right)-r\left(I\right)=r\left(\begin{array}{c}{B}_3\\ B_2\\ B_1\end{array}\right)\hfill \\ \hfill & \iff r\left(\begin{array}{ccc}I& -C_c& 0\\ I& C_c& A_1\\ 0& B_3& 0\\ 0& B_2& 0\\ 0& B_1& 0\end{array}\right)-r\left(I\right)=r\left(\begin{array}{c}{B}_3\\ B_2\\ B_1\end{array}\right)+r\left(A_1\right)\iff r\left(\begin{array}{cc}2C_c& A_1\\ B_3& 0\\ B_2& 0\\ B_1& 0\end{array}\right)=r\left(\begin{array}{c}{B}_3\\ B_2\\ B_1\end{array}\right)+r\left(A_1\right)\iff \left(12\right).\hfill \end{array}$$

Similarly, we can show that $C_{22}{L}_{B_{22}}=0\iff$ 13 and $C_{33}{L}_{B_{33}}=0\iff$ (14). Therefore, (8) holds if—and only if—(10)–(14) all hold.

We now turn our attention to show that (9) holds if—and only if—(15)–(19) hold. $R_{M_1}{C}_1{L}_{N_1}=0\iff$ (15). In fact, it follows from Lemma 2 that the following is true:

$$\begin{array}{cc}\hfill & R_{M_1}{C}_1{L}_{N_1}=0\iff r\left(R_{M_1}{C}_1{L}_{N_1}\right)=0\hfill \\ \hfill & \iff r\left(\begin{array}{cc}{C}_1& M_1\\ N_1& 0\end{array}\right)=r\left(M_1\right)+r\left(N_1\right)\hfill \\ \hfill & \iff r\left(\begin{array}{ccc}{C}_{22}& A_{22}& 0\\ B_{22}& 0& B_{11}\\ 0& A_{11}& -C_{11}\end{array}\right)=r\left(\begin{array}{c}{A}_{22}\\ A_{11}\end{array}\right)+r(B_{22},B_{11})\hfill \\ \hfill & \iff r\left(\begin{array}{cccc}{C}_c{L}_{B_2}+R_{A_1}{C}_c& A_2& 0& A_{33}\\ B_3{L}_{B_2}& 0& B_3& 0\\ 0& (I+R_{A_1})A_2& -(I+R_{A_1})C_c& 0\\ B_1{L}_{B_2}& 0& 0& 0\\ 0& 0& B_4& 0\\ 0& 0& B_2& 0\end{array}\right)\hfill \\ \hfill & =r\left(\begin{array}{cc}{B}_3{L}_{B_2}& B_3\\ B_1{L}_{B_2}& 0\\ 0& B_4\\ 0& B_2\end{array}\right)+r\left(\begin{array}{cc}{A}_2& A_{33}\\ (I+R_{A_1})A_2& 0\end{array}\right)\hfill \\ \hfill & \iff r\left(\begin{array}{ccccc}I& 0& 0& 0& 0\\ I& C_c{L}_{B_2}+R_{A_1}{C}_c& A_2& 0& A_{33}\\ 0& B_3{L}_{B_2}& 0& B_3& 0\\ I& 0& (I+R_{A_1})A_2& -(I+R_{A_1})C_c& 0\\ 0& B_1{L}_{B_2}& 0& 0& 0\\ 0& 0& 0& B_4& 0\\ 0& 0& 0& B_2& 0\end{array}\right)\hfill \\ \hfill & =r\left(\begin{array}{cc}{B}_3{L}_{B_2}& B_3\\ B_1{L}_{B_2}& 0\\ 0& B_4\\ 0& B_2\end{array}\right)+r\left(\begin{array}{ccc}I& 0& 0\\ 0& A_2& A_{33}\\ I& (I+R_{A_1})A_2& 0\end{array}\right)\hfill \\ \hfill & \iff r\left(\begin{array}{ccccccc}I& -C_c& -A_2& C_c& 0& 0& 0\\ I& C_c& A_2& 0& A_2& A_1& 0\\ 0& B_3& 0& B_3& 0& 0& 0\\ I& 0& A_2& C_c& 0& 0& A_1\\ 0& B_1& 0& 0& 0& 0& 0\\ 0& 0& 0& B_4& 0& 0& 0\\ 0& 0& 0& B_2& 0& 0& 0\\ 0& B_2& 0& 0& 0& 0& 0\end{array}\right)=r\left(\begin{array}{cc}{B}_3& B_3\\ B_1& 0\\ 0& B_4\\ 0& B_2\\ B_2& 0\end{array}\right)+r\left(\begin{array}{ccccc}I& -A_2& 0& 0& 0\\ 0& A_2& A_2& A_1& 0\\ I& A_2& 0& 0& A_1\end{array}\right)\hfill \\ \hfill & \iff r\left(\begin{array}{cccccc}2C_c& 0& 0& A_2& A_1& 0\\ B_3& 0& B_3& 0& 0& 0\\ 0& A_2& 2C_c& 0& 0& A_1\\ B_1& 0& 0& 0& 0& 0\\ 0& 0& B_1& 0& 0& 0\\ 0& 0& B_2& 0& 0& 0\\ B_2& 0& 0& 0& 0& 0\end{array}\right)=r\left(\begin{array}{cccc}{A}_2& A_1& 0& 0\\ 0& 0& A_2& A_1\end{array}\right)+r\left(\begin{array}{cc}{B}_3& B_3\\ B_2& 0\\ B_1& 0\\ 0& B_2\\ 0& B_1\end{array}\right)\iff \left(15\right).\hfill \end{array}$$

Similarly, we can show that $R_M{R}_{A}E=0\iff$ (16), $E{L}_B{L}_N=0\iff$ (17), $R_{A}E{L}_D=0\iff$ (18) and $R_{C}E{L}_B=0\iff$ (19). The proof is completed. Next, we use an Algorithm 1 for calculating Equation (1) to illustrate this theorem.    □

3.1. Algorithm with a Numerical Example

In this section, we present an algorithm and an example to illustrate Theorem 1.

Algorithm 1: Algorithm for calculating Equation (1)

(1) Feed the values of $A_i,B_j(i=1,2,j=\overline{1,3})$ and $C_c$ with conformable shapes over $\mathbb{H}$. (2) Compute the symbols in (7). (3) Check whether (8), (9) or rank equalities in (10)–(19) hold or not. If no, then return “inconsisten”. (4) Otherwise, compute $X_i(i=\overline{1,4})$.

Example 1.

Consider the matrix Equation (1). Put the following:

$$\begin{array}{cc}\hfill & A_1=\left(\begin{array}{cc}{a}_{111}& a_{112}\\ a_{121}& a_{122}\\ a_{131}& a_{132}\end{array}\right),B_1=\left(\begin{array}{ccc}{b}_{111}& b_{112}& b_{113}\\ b_{121}& b_{122}& b_{123}\end{array}\right),B_2=\left(\begin{array}{ccc}{b}_{211}& b_{212}& b_{213}\\ b_{221}& b_{222}& b_{223}\end{array}\right),\hfill \\ \hfill & A_2=\left(\begin{array}{cc}{a}_{211}& a_{212}\\ a_{221}& a_{222}\\ a_{231}& a_{232}\end{array}\right),B_3=\left(\begin{array}{ccc}{b}_{311}& b_{312}& b_{313}\\ b_{321}& b_{322}& b_{323}\end{array}\right),C_c=\left(\begin{array}{ccc}{c}_{11}& c_{12}& c_{13}\\ c_{21}& c_{22}& c_{23}\\ c_{31}& c_{32}& c_{33}\end{array}\right),\hfill \end{array}$$

where

$$\begin{array}{cc}\hfill & a_{111}=0.3181+0.5447\mathbf{i}+0.2187\mathbf{j}+0.3685\mathbf{k},a_{112}=0.6456+0.7210\mathbf{i}+0.0636\mathbf{j}+0.7720\mathbf{k},\hfill \\ \hfill & a_{121}=0.1192+0.6473\mathbf{i}+0.1058\mathbf{j}+0.7635\mathbf{k},a_{122}=0.4795+0.5225\mathbf{i}+0.4046\mathbf{j}+0.9329\mathbf{k},\hfill \\ \hfill & a_{131}=0.9398+0.5439\mathbf{i}+0.1097\mathbf{j}+0.6279\mathbf{k},a_{132}=0.6393+0.9937\mathbf{i}+0.4484\mathbf{j}+0.9727\mathbf{k},\hfill \\ \hfill & b_{111}=0.1920+0.8611\mathbf{i}+0.3477\mathbf{j}+0.2428\mathbf{k},b_{112}=0.6963+0.3935\mathbf{i}+0.5861\mathbf{j}+0.6878\mathbf{k},\hfill \\ \hfill & b_{113}=0.5254+0.7413\mathbf{i}+0.0445\mathbf{j}+0.7363\mathbf{k},b_{121}=0.1389+0.4849\mathbf{i}+0.1500\mathbf{j}+0.4424\mathbf{k},\hfill \\ \hfill & b_{122}=0.0938+0.6714\mathbf{i}+0.2621\mathbf{j}+0.3592\mathbf{k},b_{123}=0.5303+0.5201\mathbf{i}+0.7549\mathbf{j}+0.3947\mathbf{k},\hfill \\ \hfill & b_{211}=0.6834+0.2703\mathbf{i}+0.7691\mathbf{j}+0.7904\mathbf{k},b_{212}=0.4423+0.8217\mathbf{i}+0.8085\mathbf{j}+0.3276\mathbf{k},\hfill \\ \hfill & b_{213}=0.3309+0.8878\mathbf{i}+0.3774\mathbf{j}+0.4386\mathbf{k},b_{221}=0.7040+0.1971\mathbf{i}+0.3968\mathbf{j}+0.9493\mathbf{k},\hfill \\ & b_{222}=0.0196+0.4299\mathbf{i}+0.7551\mathbf{j}+0.6713\mathbf{k},b_{223}=0.4243+0.3912\mathbf{i}+0.2160\mathbf{j}+0.8335\mathbf{k},\hfill \\ \hfill & a_{211}=0.7689+0.5880\mathbf{i}+0.7900\mathbf{j}+0.6787\mathbf{k},a_{212}=0.9899+0.4070\mathbf{i}+0.0900\mathbf{j}+0.4950\mathbf{k},\hfill \\ \hfill & a_{221}=0.1673+0.1548\mathbf{i}+0.3185\mathbf{j}+0.4952\mathbf{k},a_{222}=0.5144+0.7487\mathbf{i}+0.1117\mathbf{j}+0.1476\mathbf{k},\hfill \\ \hfill & a_{231}=0.8620+0.1999\mathbf{i}+0.5341\mathbf{j}+0.1897\mathbf{k},a_{232}=0.8843+0.8256\mathbf{i}+0.1363\mathbf{j}+0.0550\mathbf{k},\hfill \\ \hfill & b_{311}=0.8507+0.8790\mathbf{i}+0.5277\mathbf{j}+0.5747\mathbf{k},b_{312}=0.9296+0.0005\mathbf{i}+0.8013\mathbf{j}+0.7386\mathbf{k},\hfill \\ \hfill & b_{313}=0.5828+0.6126\mathbf{i}+0.4981\mathbf{j}+0.2467\mathbf{k},b_{321}=0.5606+0.9889\mathbf{i}+0.4795\mathbf{j}+0.8452\mathbf{k},\hfill \\ & b_{322}=0.6967+0.8654\mathbf{i}+0.2278\mathbf{j}+0.5860\mathbf{k},b_{323}=0.8154+0.9900\mathbf{i}+0.9009\mathbf{j}+0.6664\mathbf{k},\hfill \\ \hfill & c_{11}=-38.7863+4.0617\mathbf{i}+0.5536\mathbf{j}-3.4984\mathbf{k},c_{12}=-35.3609-9.0836\mathbf{i}-1.7527\mathbf{j}-6.7689\mathbf{k},\hfill \\ \hfill & c_{13}=-33.503-3.7707\mathbf{i}+5.2872\mathbf{j}-6.2708\mathbf{k},c_{21}=-24.7749-3.6921\mathbf{i}-0.5143\mathbf{j}-10.8211\mathbf{k},\hfill \\ \hfill & c_{22}=-21.0950-11.3075\mathbf{i}-3.9522\mathbf{j}-10.8211\mathbf{k},c_{23}=-18.9376-7.7280\mathbf{i}+1.7134\mathbf{j}-11.9519\mathbf{k},\hfill \\ \hfill & c_{31}=-36.2182+5.3877\mathbf{i}-2.8839\mathbf{j}-0.8389\mathbf{k},c_{32}=-33.9232-7.3248\mathbf{i}-4.3221\mathbf{j}-3.9587\mathbf{k},\hfill \\ \hfill & c_{33}=-31.8026-1.6492\mathbf{i}+1.2342\mathbf{j}-6.7683\mathbf{k}.\hfill \end{array}$$

Computing directly yields the following:

$$\begin{array}{cc}\hfill & r\left(\begin{array}{ccc}2C_c& A_2& A_1\\ B_1& 0& 0\\ B_2& 0& 0\end{array}\right)=r(A_2,A_1)+r\left(\begin{array}{c}{B}_1\\ B_2\end{array}\right)=6,\hfill \\ \hfill & r\left(\begin{array}{ccc}2C_c& A_2& A_1\\ B_2& 0& 0\end{array}\right)=r(A_2,A_1)+r\left(B_2\right)=5,\hfill \\ \hfill & r\left(\begin{array}{cc}2C_c& A_1\\ B_3& 0\\ B_2& 0\\ B_1& 0\end{array}\right)=r\left(A_1\right)+r\left(\begin{array}{c}{B}_3\\ B_2\\ B_1\end{array}\right)=5,\hfill \end{array}$$

$$\begin{array}{cc}\hfill & r\left(\begin{array}{ccc}2C_c& A_2& A_1\\ B_3& 0& 0\\ B_2& 0& 0\\ B_1& 0& 0\end{array}\right)=r(A_2,A_1)+r\left(\begin{array}{c}{B}_3\\ B_2\\ B_1\end{array}\right)=6,\hfill \\ \hfill & r\left(\begin{array}{cc}{C}_c& A_1\\ B_3& 0\\ B_2& 0\end{array}\right)=r\left(A_1\right)+r\left(\begin{array}{c}{B}_3\\ B_2\end{array}\right)=5,\hfill \\ \hfill & r\left(\begin{array}{cccccc}2C_c& 0& 0& A_2& A_1& 0\\ B_3& 0& B_3& 0& 0& 0\\ 0& A_2& -2C_c& 0& 0& A_1\\ B_1& 0& 0& 0& 0& 0\\ 0& 0& B_1& 0& 0& 0\\ 0& 0& B_2& 0& 0& 0\\ B_2& 0& 0& 0& 0& 0\end{array}\right)=r\left(\begin{array}{cccc}{A}_2& A_1& 0& 0\\ 0& 0& A_2& A_1\end{array}\right)+r\left(\begin{array}{cc}{B}_3& B_3\\ B_2& 0\\ B_1& 0\\ 0& B_2\\ 0& B_1\end{array}\right)=12,\hfill \\ \hfill & r\left(\begin{array}{ccccc}0& B_3& B_3& 0& 0\\ -2A_2& 2C_c& 0& A_1& 0\\ A_2& 0& C_c& 0& A_1\\ 0& B_2& 0& 0& 0\\ 0& B_1& 0& 0& 0\\ 0& 0& B_2& 0& 0\end{array}\right)=r\left(\begin{array}{ccc}0& A_1& 0\\ A_2& 0& A_1\end{array}\right)+r\left(\begin{array}{cc}{B}_3& B_3\\ B_2& 0\\ B_1& 0\\ 0& B_2\end{array}\right)=11,\hfill \\ \hfill & r\left(\begin{array}{ccccccccc}0& 0& -B_3& B_3& B_3& 0& 0& 0& 0\\ 0& 0& 2C_c& 0& 0& A_2& 0& A_1& 0\\ A_2& 0& 0& 2C_c& 0& 0& 0& 0& A_1\\ 0& A_2& 0& 0& C_c& 0& A_1& 0& 0\\ 0& 0& 0& 0& B_2& 0& 0& 0& 0\\ 0& 0& B_1& 0& 0& 0& 0& 0& 0\\ 0& 0& B_2& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& B_2& 0& 0& 0& 0& 0\\ 0& 0& 0& B_1& 0& 0& 0& 0& 0\end{array}\right)\hfill \\ \hfill & =r\left(\begin{array}{cccccc}{A}_2& A_1& 0& 0& 0& 0\\ 0& 0& A_2& A_1& 0& 0\\ 0& 0& 0& 0& A_2& A_1\end{array}\right)+r\left(\begin{array}{ccc}{B}_3& B_3& B_3\\ B_2& 0& 0\\ B_1& 0& 0\\ 0& B_2& 0\\ 0& B_1& 0\\ 0& 0& B_2\end{array}\right)=18,\hfill \end{array}$$

$$\begin{array}{cc}\hfill & r\left(\begin{array}{cccccccc}0& B_3& B_3& 0& 0& 0& 0& 0\\ 0& 0& -B_3& B_3& 0& 0& 0& 0\\ 2A_2& 0& 2C_c& 0& 0& 0& A_1& 0\\ 0& 2C_c& 0& 0& 0& A_2& 0& A_1\\ A_2& 0& 0& C_c& A_1& 0& 0& 0\\ 0& B_2& 0& 0& 0& 0& 0& 0\\ 0& B_1& 0& 0& 0& 0& 0& 0\\ 0& 0& B_2& 0& 0& 0& 0& 0\\ 0& 0& B_1& 0& 0& 0& 0& 0\\ 0& 0& 0& B_2& 0& 0& 0& 0\end{array}\right)\hfill \\ \hfill & =r\left(\begin{array}{ccccc}0& A_2& A_1& 0& 0\\ A_2& 0& 0& A_1& 0\\ A_2& 0& 0& 0& A_1\end{array}\right)+r\left(\begin{array}{ccc}{B}_3& 0& 0\\ B_3& B_3& 0\\ 0& B_3& B_3\\ B_2& 0& 0\\ B_1& 0& 0\\ 0& B_2& 0\\ 0& B_1& 0\\ 0& 0& B_2\end{array}\right)=17,\hfill \end{array}$$

$$\begin{array}{cc}\hfill & r\left(\begin{array}{ccccccccc}0& 0& B_3& B_3& B_3& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& B_3& 0& 0& 0\\ 0& 0& 2C_c& 0& 0& 0& A_1& A_2& 0\\ A_2& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& A_1\\ 0& A_2& 0& 0& 0& C_c& 0& 0& 0\\ 0& 0& B_1& 0& 0& 0& 0& 0& 0\\ 0& 0& B_2& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& B_1& 0& 0& 0& 0& 0\\ 0& 0& 0& B_2& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& B_1& 0& 0& 0& 0\\ 0& 0& 0& 0& B_2& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& B_2& 0& 0& 0\end{array}\right)\hfill \\ \hfill & =r\left(\begin{array}{ccccc}0& 0& A_2& A_1& 0\\ A_2& 0& 0& 0& 0\\ 0& 0& 0& 0& A_1\\ 0& A_2& 0& 0& 0\end{array}\right)+r\left(\begin{array}{cccc}{B}_3& B_3& B_3& 0\\ 0& 0& 0& B_3\\ B_2& 0& 0& 0\\ B_1& 0& 0& 0\\ 0& B_2& 0& 0\\ 0& B_1& 0& 0\\ 0& 0& B_1& 0\\ 0& 0& B_2& 0\\ 0& 0& 0& B_2\end{array}\right)=24.\hfill \end{array}$$

All rank equalities in (10)–(19) hold. Hence, according to Theorem 1, Equation (1) is consistent, and the general solution to the matrix Equation (1) can be expressed as follows:

$$\begin{array}{cc}\hfill & X_1=\left(\begin{array}{cc}0.500+3.0117\mathbf{i}+0.0161\mathbf{j}+1.2813\mathbf{k}& -2.2870-5.3594\mathbf{i}+2.8008\mathbf{j}-3.4531\mathbf{k}\\ -0.3281+0.1484\mathbf{i}+1.4375\mathbf{j}-1.1836\mathbf{k}& 2.2500+1.9219\mathbf{i}+0.4844\mathbf{j}+7.1406\mathbf{k}\end{array}\right)+\hfill \\ \hfill & V_2\left(\begin{array}{cc}0.2734-0.0010\mathbf{i}-0.0020\mathbf{j}-0.0078\mathbf{k}& -0.0117-0.1328\mathbf{i}-0.0625\mathbf{j}-0.4063\mathbf{k}\\ 0.0508+0.1519\mathbf{i}+0.1387\mathbf{j}+0.4042\mathbf{k}& -0.1973+0.0039\mathbf{i}-0.2617\mathbf{j}\end{array}\right),\hfill \\ \hfill & X_2=\left(\begin{array}{cc}3\mathbf{i}+2\mathbf{j}& 2\mathbf{k}+\mathbf{i}\\ 2\mathbf{j}+3\mathbf{i}& 2\mathbf{i}+\mathbf{j}\end{array}\right),X_3=\left(\begin{array}{cc}{x}_{311}& x_{312}\\ x_{321}& x_{322}\end{array}\right),X_4=\left(\begin{array}{cc}{x}_{0411}& x_{0412}\\ x_{0421}& x_{0422}\end{array}\right),\hfill \end{array}$$

or

$$\begin{array}{cc}\hfill & X_4=\left(\begin{array}{cc}{x}_{411}& x_{412}\\ x_{421}& x_{422}\end{array}\right)+T_1{U}_5+U_6{T}_2,U_1=S_1(U_{11}-U_{12}{T}_7{U}_{13}+U_{14}{T}_6),\hfill \\ \hfill & U_2=\left\{\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right)T_7+T_8\left(\begin{array}{cccc}{u}_{211}& u_{212}& u_{213}& u_{214}\\ u_{221}& u_{222}& u_{223}& u_{224}\\ u_{231}& u_{232}& u_{233}& u_{234}\\ u_{241}& u_{242}& u_{243}& u_{244}\end{array}\right)\right\}{S}_2,\hfill \\ \hfill & U_3=\left(\begin{array}{cc}{u}_{311}& u_{312}\\ u_{321}& u_{322}\end{array}\right),U_4=\left(\begin{array}{cc}{u}_{411}& u_{412}\\ u_{421}& u_{422}\end{array}\right),U_5=-S_3(U_{11}-U_{12}{T}_7{U}_{13}+U_{14}{T}_6),\hfill \\ \hfill & U_6=-\left\{\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right)T_7+T_8\left(\begin{array}{cccc}{u}_{211}& u_{212}& u_{213}& u_{214}\\ u_{221}& u_{222}& u_{223}& u_{224}\\ u_{231}& u_{232}& u_{233}& u_{234}\\ u_{241}& u_{242}& u_{243}& u_{244}\end{array}\right)\right\}{S}_4,U_{14}=\left(\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right),\hfill \\ \hfill & T_1=\left(\begin{array}{cc}0.2500+0.0447\mathbf{i}-0.0938\mathbf{j}-0.2090\mathbf{k}& -0.1348+0.3086\mathbf{i}-0.0781\mathbf{j}+0.2188\mathbf{k}\\ -0.0938-0.3564\mathbf{i}+0.0313\mathbf{j}+0.3167\mathbf{k}& 0.4102-0.0117\mathbf{i}+0.1094\mathbf{j}-0.2109\mathbf{k}\end{array}\right),\hfill \\ \hfill & T_2=\left(\begin{array}{cc}0.0005+0.0012\mathbf{i}+0.0005\mathbf{j}+0.0005\mathbf{k}& -0.0002+0.0005\mathbf{i}+0.0005\mathbf{j}-0.0005\mathbf{k}\\ -0.0005-0.0004\mathbf{i}-0.0010\mathbf{j}+0.0010\mathbf{k}& 0.0012\mathbf{i}+0.0020\mathbf{j}+0.0002\mathbf{k},\end{array}\right),\hfill \end{array}$$

$$\begin{array}{cc}\hfill & U_{11}=\left(\begin{array}{cc}0& 0\\ 0& 0\\ 0.2157-0.5625\mathbf{i}-0.5079\mathbf{j}+0.4975\mathbf{k}& 0.0479+0.8423\mathbf{i}+1.4991\mathbf{j}+0.0312\mathbf{k}\\ 0.4770+2.4543\mathbf{i}+2.5170\mathbf{j}+2.3890\mathbf{k}& -2.9781-2.4611\mathbf{i}-0.1109\mathbf{j}-4.8836\mathbf{k}\end{array}\right),\hfill \\ \hfill & U_{12}=\left(\begin{array}{cc}0& 0\\ 0& 0\\ 2.0941-1.8169\mathbf{i}-19.7767\mathbf{j}+7.1464\mathbf{k}& 4.0003-7.6426\mathbf{i}-23.6916\mathbf{j}-12.6004\mathbf{k}\\ 35.0358+22.9322\mathbf{i}-19.0766\mathbf{j}+62.5683\mathbf{k}& 50.3227+10.8273\mathbf{i}-41.1511\mathbf{j}+44.4639\mathbf{k}\end{array}\right),\hfill \\ \hfill & U_{13}=\left(\begin{array}{cc}0& 0\\ 0& 0\\ 0.0005+0.0012\mathbf{i}+0.0005\mathbf{k}& -0.0002+0.0005\mathbf{i}+0.0005\mathbf{j}+0.0005\mathbf{k}\\ -0.0005-0.0004\mathbf{i}-0.0010\mathbf{j}+0.0010\mathbf{k}& 0.0012\mathbf{i}+0.0020\mathbf{j}+0.0002\mathbf{k}\end{array}\right),\hfill \end{array}$$

where

$$\begin{array}{cc}\hfill & x_{311}=2.4249\times {10}^{15}+9.158\times {10}^{14}\mathbf{i}+3.6562\times {10}^{15}\mathbf{j}-9.802\times {10}^{14}\mathbf{k},\hfill \\ \hfill & x_{312}=2.8915\times {10}^{15}-3.424\times {10}^{15}\mathbf{i}+3.6682\times {10}^{15}\mathbf{j}+3.8670\times {10}^{14}\mathbf{k},\hfill \\ \hfill & x_{321}=-2.8900\times {10}^{15}-8.349\times {10}^{14}\mathbf{i}+4.8121\times {10}^{15}\mathbf{j}+1.3028\times {10}^{15}\mathbf{k},\hfill \\ \hfill & x_{322}=-2.876\times {10}^{15}+3.0166\times {10}^{15}\mathbf{i}+1.3349\times {10}^{15}\mathbf{j}+5.6492\times {10}^{14}\mathbf{k},\hfill \\ \hfill & x_{0411}=-1.0460\times {10}^{14}-5.5967\times {10}^{14}\mathbf{i}-4.5730\times {10}^{14}\mathbf{j}+2.5023\times {10}^{14}\mathbf{k},\hfill \\ \hfill & x_{0412}=4.0206\times {10}^{14}+1.7870\times {10}^{14}\mathbf{i}+3.5717\times {10}^{14}\mathbf{j}+1.6691\times {10}^{14}\mathbf{k},\hfill \\ \hfill & x_{0421}=-1.0061\times {10}^{14}+3.6673\times {10}^{14}\mathbf{i}+4.9065\times {10}^{14}\mathbf{j}-7.1850\times {10}^{14}\mathbf{k},\hfill \\ \hfill & x_{0411}=-1.0460\times {10}^{14}-5.5967\times {10}^{14}\mathbf{i}-4.5730\times {10}^{14}\mathbf{j}+2.5023\times {10}^{14}\mathbf{k},\hfill \\ \hfill & x_{0422}=-2.0371\times {10}^{14}-1.9449\times {10}^{14}\mathbf{i}-6.7917\times {10}^{14}\mathbf{j}+3.27\times {10}^{11}\mathbf{k},\hfill \\ \hfill & x_{412}=-2.5081\times {10}^{15}-2.526\times {10}^{14}\mathbf{i}-1.8086\times {10}^{15}\mathbf{j}-1.6121\times {10}^{15}\mathbf{k},\hfill \\ \hfill & x_{421}=-1.6826\times {10}^{15}+3.233\times {10}^{14}\mathbf{i}+1.2608\times {10}^{14}\mathbf{j}-5.905\times {10}^{14}\mathbf{k},\hfill \\ \hfill & x_{422}=3.376\times {10}^{14}-2.1855\times {10}^{15}\mathbf{i}-2.5144\times {10}^{15}\mathbf{j}-9.287\times {10}^{14}\mathbf{k},\hfill \\ \hfill & u_{311}=-1.093\times {10}^{35}-9.730\times {10}^{35}\mathbf{i}+1.8576\times {10}^{36}\mathbf{j}-1.9289\times {10}^{36}\mathbf{k},\hfill \\ \hfill & u_{312}=8.155\times {10}^{35}+9.72\times {10}^{34}\mathbf{i}-1.5978\times {10}^{36}\mathbf{j}-1.5691\times {10}^{36}\mathbf{k},\hfill \\ \hfill & u_{321}=-8.932\times {10}^{35}-1.8508\times {10}^{36}\mathbf{i}+4.0088\times {10}^{36}\mathbf{j}-4.8557\times {10}^{36}\mathbf{k},\hfill \\ \hfill & u_{322}=1.5025\times {10}^{36}-7.623\times {10}^{35}\mathbf{i}-4.0296\times {10}^{36}\mathbf{j}-3.3906\times {10}^{36}\mathbf{k},\hfill \\ \hfill & u_{411}=1.4764\times {10}^{35}+4.669\times {10}^{34}\mathbf{i}-5.229\times {10}^{34}\mathbf{j}-2.3157\times {10}^{35}\mathbf{k},\hfill \\ \hfill & u_{412}=-3.7703\times {10}^{35}+2.5079\times {10}^{35}\mathbf{i}-4.3196\times {10}^{35}\mathbf{j}+5.1752\times {10}^{35}\mathbf{k},\hfill \\ \hfill & u_{421}=1.3750\times {10}^{35}-1.2281\times {10}^{35}\mathbf{i}-2.4962\times {10}^{35}\mathbf{j}+1.7278\times {10}^{35}\mathbf{k},\hfill \\ \hfill & u_{422}=4.310\times {10}^{34}+5.1989\times {10}^{35}\mathbf{i}+8.0931\times {10}^{35}\mathbf{j}+2.5832\times {10}^{35}\mathbf{k},\hfill \\ \hfill & u_{211}=0.9999+0.0001\mathbf{j}-0.0001\mathbf{k},u_{212}=-0.0002+0.0001\mathbf{i}+0.0004\mathbf{j},\hfill \\ \hfill & u_{213}=0.0025-0.0052\mathbf{i}-0.0085\mathbf{j}-0.0084\mathbf{k},u_{214}=0.0052-0.0006\mathbf{i}-0.0064\mathbf{j}+0.0067\mathbf{k},\hfill \\ \hfill & u_{221}=-0.0002-0.0001\mathbf{i}-0.0001\mathbf{j}-0.0002\mathbf{k},u_{222}=0.9994+0.0001\mathbf{i}-0.0005\mathbf{k},\hfill \\ \hfill & u_{223}=0.0096-0.0130\mathbf{i}-0.0134\mathbf{j}+0.0039\mathbf{k},u_{224}=0.0098+0.0104\mathbf{i}+0.0081\mathbf{j}+0.0108\mathbf{k},\hfill \\ \hfill & u_{231}=0.0025+0.0052\mathbf{i}-0.0002\mathbf{j}-0.0013\mathbf{k},u_{232}=0.0096+0.0130\mathbf{i}-0.0014\mathbf{j}-0.0031\mathbf{k},\hfill \\ \hfill & u_{233}=0.0006+0.0001\mathbf{i}-0.0001\mathbf{k},u_{234}=-0.0001+0.0005\mathbf{i}-0.0001\mathbf{j},\hfill \\ \hfill & u_{241}=0.0052+0.0006\mathbf{i}-0.0040\mathbf{j}-0.0043\mathbf{k},u_{242}=0.0098-0.0104\mathbf{i}+0.0036\mathbf{j}-0.0096\mathbf{k},\hfill \\ \hfill & u_{243}=-0.0001-0.0005\mathbf{i}+0.0004\mathbf{j}-0.0001\mathbf{k},u_{244}=0.0005-0.0002\mathbf{j}-0.0003\mathbf{k},\hfill \end{array}$$

$V_2$ is an arbitrary matrix of order $2\times 2$ over quaternion $\mathbb{H}$, $T_6$ is an arbitrary matrix of order $4\times 2$ over quaternion $\mathbb{H}$, and $T_7$ and $T_8$ are arbitrary matrix of order $2\times 4$ over quaternion $\mathbb{H}$, $S_1=\left(\begin{array}{cc}{I}_2& 0\end{array}\right),S_2=\left(\begin{array}{c}{I}_2\\ 0\end{array}\right)$, $S_3=\left(\begin{array}{cc}0& I_2\end{array}\right),S_4=\left(\begin{array}{c}0\\ I_2\end{array}\right)$.

3.2. The General Solution to the System (4)

Based on Theorem 1, in this section, we consider the system (4) with the known matrices $E_1,E_2$, $E_{11},E_{22}$, $F_i,H_i(i=\overline{1,4})$, $G_j,F_{jj}(j=\overline{1,3})$, and T over $\mathbb{H}$. For convenience, we define the notation as follows:

$$\begin{array}{cc}\hfill & A_i=E_{ii}{L}_{E_i},B_j=R_{G_j}{F}_{jj}(i=1,2,j=\overline{1,3}),C_c=T-E_{11}(E_1^{-}{F}_1+L_{E_1}{H}_1{G}_1^{-})F_{11}\hfill \\ \hfill & -E_{11}(E_1^{-}{F}_2+L_{E_1}{H}_2{G}_2^{-})F_{22}-E_{22}(E_2^{-}{F}_3+L_{E_2}{H}_3{G}_2^{-})F_{22}-E_{22}(E_1^{-}{F}_4+L_{E_2}{H}_4{G}_3^{-})F_{33},\hfill \\ \hfill & B_4=B_1{L}_{B_2},B_5=B_2{L}_{B_4},A_{33}=R_{A_1}{A}_2,B_{33}=B_3{L}_{B_2},\hfill \\ \hfill & C_{33}=R_{A_1}{C}_c{L}_{B_2},A_{11}=A_2+A_{33},C_{44}=C_c{L}_{B_2}+R_{A_1}{C}_c,B_{11}=B_3{L}_{B_4}{L}_{B_5},\hfill \\ & C_{11}=(I+R_{A_1})C_c{L}_{B_4}{L}_{B_5},A_{22}=R_{A_{33}}{A}_2,B_{22}=B_3{L}_{B_2}{L}_{B_4},C_{22}=R_{A_{33}}{C}_{44}{L}_{B_4},\hfill \\ & M_1=A_{22}{L}_{A_{11}},N_1=R_{B_{11}}{B}_{22},C_1=C_{22}-A_{22}{A}_{11}^{-}{C}_{11}{B}_{11}^{-}{B}_{22},D_1=R_{A_1}{A}_{22},\hfill \\ \hfill & \varphi =A_{11}^{-}{C}_{11}{B}_{11}^{-}+L_{A_{11}}{M}_1^{-}{C}_1{B}_{22}^{-}-L_{A_{11}}{M}_1^{-}{A}_{22}{D}_1^{-}{R}_{M_1}{C}_1{B}_{22}^{-}+D_1^{-}{R}_{M_1}{C}_1{N}_1^{-}{R}_{B_{11}},\hfill \\ \hfill & M_2=\left[L_{A_{11}}{L}_{M_1},L_{A_{33}}\right],N_2=\left[\begin{array}{c}{R}_{N_1}{R}_{B_{11}}\\ R_{B_{33}}\end{array}\right],A=R_{M_2}{L}_{A_{11}},B=R_{B_{22}}{L}_{N_2},C=R_{M_2}{L}_{A_{22}},\hfill \\ \hfill & D=R_{B_{11}}{L}_{N_2},E_1=A_{33}^{-}{C}_{33}{B}_{33}^{-}-\varphi ,E=R_{M_2}{E}_1{L}_{N_2},M=R_{A}C,N=D{L}_B,S=C{L}_M,\hfill \end{array}$$

(34)

and

$$S_1=\left[I_m,0\right],S_2=\left[\begin{array}{c}{I}_n\\ 0\end{array}\right],S_3=\left[0,I_m\right],S_4=\left[\begin{array}{c}0\\ I_n\end{array}\right].$$

Thus, we obtain the following—one of the main result of this paper.

Theorem 2.

Consider the system of quaternion matrix Equation (4) with the notation given in (34). The following statements are equivalent:

(1)

The system of matrix Equation (4) is consistent.

(2)

$$\begin{array}{c}\hfill E_1{H}_1=F_1{G}_1,E_1{H}_2=F_2{G}_2,E_2{H}_3=F_3{G}_2,E_2{H}_4=F_4{G}_3\end{array}$$

(35)

and

$$\begin{array}{cc}\hfill & \begin{array}{c}\hfill R_{E_1}{F}_1=0,R_{E_2}{F}_3=0,H_1{L}_{G_1}=0,H_2{L}_{G_2}=0,H_4{L}_{G_3}=0,\end{array}\hfill \end{array}$$

(36)

$$\begin{array}{cc}\hfill & \begin{array}{cc}\hfill & R_{A_{33}}{R}_{A_1}{C}_{44}=0,R_{A_{11}}{C}_{11}=0,C_{ii}{L}_{B_{ii}}=0(i=\overline{1,3}),R_{M_1}{C}_1{L}_{N_1}=0,\hfill \\ \hfill & R_M{R}_{A}E=0,E{L}_B{L}_N=0,R_{A}E{L}_D=0,R_{C}E{L}_B=0.\hfill \end{array}\hfill \end{array}$$

(37)

(3)

(35) holds and

$$\begin{array}{cc}\hfill & \begin{array}{cc}\hfill & r\left(\begin{array}{cc}{F}_1& E_1\end{array}\right)=r\left(E_1\right),r\left(\begin{array}{cc}{F}_3& E_2\end{array}\right)=r\left(E_2\right),\hfill \\ \hfill & r\left(\begin{array}{c}{H}_1\\ G_1\end{array}\right)=r\left(G_1\right),r\left(\begin{array}{c}{H}_2\\ G_2\end{array}\right)=r\left(G_2\right),r\left(\begin{array}{c}{H}_4\\ G_3\end{array}\right)=r\left(G_3\right),\hfill \end{array}\hfill \end{array}$$

(38)

$$\begin{array}{cc}\hfill & \begin{array}{cc}\hfill & r\left(\begin{array}{cc}{P}_i& S_i\\ 0& Q_i\end{array}\right)=r\left(P_i\right)+r\left(Q_i\right),i=\overline{1,10}\hfill \end{array}\hfill \end{array}$$

(39)

where

$$\begin{array}{cc}\hfill & \begin{array}{cc}\hfill & P_1=\left(\begin{array}{cc}{E}_{22}& E_{11}\\ E_2& 0\\ 0& E_1\end{array}\right),Q_1=\left(\begin{array}{cc}{F}_{22}& G_2\end{array}\right),S_1=\left(\begin{array}{cc}2T& 0\\ 2W_2& 0\\ 2W_1& 0\end{array}\right),P_2=\left(\begin{array}{cc}{E}_{22}& E_{11}\\ E_2& 0\\ 0& E_1\end{array}\right),\hfill \end{array}\hfill \end{array}$$

(40)

$$\begin{array}{cc}\hfill & \begin{array}{cc}\hfill & Q_2=\left(\begin{array}{ccc}{F}_{11}& G_1& 0\\ F_{22}& 0& G_2\end{array}\right),S_2=\left(\begin{array}{ccc}2T& 0& 2V_1\\ 2W_2& 0& 0\\ 2U_1& 0& 0\end{array}\right),Q_3=\left(\begin{array}{cccc}{F}_{33}& G_3& 0& 0\\ F_{22}& 0& G_2& 0\\ F_{11}& 0& 0& G_1\end{array}\right),\hfill \end{array}\hfill \end{array}$$

(41)

$$\begin{array}{cc}\hfill & \begin{array}{cc}\hfill & P_3=\left(\begin{array}{c}{E}_{11}\\ E_1\end{array}\right),S_3=\left(\begin{array}{cccc}2T& 2V_2& 2V_3& 2V_4\\ 2W_2& 0& 0\\ 2U_2& 0& 0& 0\end{array}\right),S_4=\left(\begin{array}{cccc}2T& 2V_2& 2V_1& 0\\ 2U_3& 0& 0& 0\\ 2U_1& 0& 0& 0\end{array}\right),\hfill \end{array}\hfill \end{array}$$

(42)

$$\begin{array}{cc}\hfill & \begin{array}{cc}\hfill & P_4=\left(\begin{array}{cc}{E}_{22}& E_{11}\\ E_2& 0\\ 0& E_1\end{array}\right),Q_4=\left(\begin{array}{cccc}{F}_{33}& G_3& 0& 0\\ F_{22}& 0& G_2& 0\\ F_{11}& 0& 0& G_1\end{array}\right),Q_5=\left(\begin{array}{ccc}{F}_{33}& G_3& 0\\ F_{22}& 0& G_2\end{array}\right),\hfill \end{array}\hfill \end{array}$$

(43)

$$\begin{array}{cc}\hfill & \begin{array}{cc}\hfill & P_5=\left(\begin{array}{c}{E}_{11}\\ E_1\end{array}\right),S_5=\left(\begin{array}{ccc}T& V_2& V_3\\ W_1& 0& 0\end{array}\right),Q_6=\left(\begin{array}{ccccccc}{F}_{33}& F_{33}& G_3& 0& 0& 0& 0\\ F_{22}& 0& 0& G_2& 0& 0& 0\\ F_{11}& 0& 0& 0& G_1& 0& 0\\ 0& F_{22}& 0& 0& 0& G_2& 0\\ 0& F_{11}& 0& 0& 0& 0& G_1\end{array}\right),\hfill \end{array}\hfill \end{array}$$

(44)

$$\begin{array}{cc}\hfill & \begin{array}{cc}\hfill & P_6=\left(\begin{array}{cccc}{E}_{22}& E_{11}& 0& 0\\ 0& 0& E_{22}& E_{11}\\ E_2& 0& 0& 0\\ 0& E_1& 0& 0\\ 0& 0& E_2& 0\\ 0& 0& 0& E_1\end{array}\right),S_6=\left(\begin{array}{ccccccc}2T& 0& 0& 2V_3& 2V_4& 0& 0\\ 0& 2T& 0& 0& 0& 2V_3& 2V_4\\ 2U_4& 0& 0& 0& 0& 0& 0\\ 2U_5& 0& 0& 0& 0& 0& 0\\ 0& 2U_4& 0& 0& 0& 0& 0\\ 0& 2U_5& 0& 0& 0& 0& 0\end{array}\right),\hfill \end{array}\hfill \end{array}$$

$$\begin{array}{cc}\hfill & \begin{array}{cc}\hfill & P_7=\left(\begin{array}{cccccc}{E}_{22}& E_{11}& 0& 0& 0& 0\\ 0& 0& E_{22}& E_{11}& 0& 0\\ 0& 0& 0& 0& E_{22}& E_{11}\\ E_2& 0& 0& 0& 0& 0\\ 0& E_1& 0& 0& 0& 0\\ 0& 0& E_2& 0& 0& 0\\ 0& 0& 0& E_1& 0& 0\\ 0& 0& 0& 0& E_2& 0\\ 0& 0& 0& 0& 0& E_1\end{array}\right),S_7=\left(\begin{array}{ccccccc}2T& 0& 0& 2V_3& 2V_4& 0& 0\\ 0& 2T& 0& 0& 0& 2V_3& 2V_4\\ 2U_4& 0& 0& 0& 0& 0& 0\\ 2U_5& 0& 0& 0& 0& 0& 0\\ 0& 2U_4& 0& 0& 0& 0& 0\\ 0& 2U_5& 0& 0& 0& 0& 0\end{array}\right),\hfill \end{array}\hfill \end{array}$$

(45)

$$\begin{array}{cc}\hfill & \begin{array}{cc}\hfill & Q_7=\left(\begin{array}{ccccccccc}{F}_{33}& F_{33}& F_{33}& G_3& 0& 0& 0& 0& 0\\ F_{22}& 0& 0& 0& G_2& 0& 0& 0& 0\\ F_{11}& 0& 0& 0& 0& G_1& 0& 0& 0\\ 0& F_{22}& 0& 0& 0& 0& G_2& 0& 0\\ 0& F_{11}& 0& 0& 0& 0& 0& G_1& 0\\ 0& 0& F_{22}& 0& 0& 0& 0& 0& G_2\end{array}\right),P_8=\left(\begin{array}{ccccc}2E_{22}& 0& E_{11}& 0& 0\\ E_{22}& E_{11}& 0& 0& 0\\ 0& 0& 0& E_{22}& E_{11}\\ E_2& 0& 0& 0& 0\\ 0& E_1& 0& 0& 0\\ 0& 0& E_1& 0& 0\\ 0& 0& 0& E_2& 0\\ 0& 0& 0& 0& E_1\end{array}\right),\hfill \end{array}\hfill \end{array}$$

(46)

$$\begin{array}{cc}\hfill & \begin{array}{cc}\hfill & S_8=\left(\begin{array}{cccccccccc}0& 2T& 0& 0& 0& 0& 0& 2V_3& 2V_4& 0\\ T& 0& T& 0& 0& V_3& V_4& 0& 0& V_3\\ 2T& 0& 0& 0& 0& 2V_3& 2V_4& 0& 0& 0\\ 2U_4& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ U_4& 0& U_4& 0& 0& 0& 0& 0& 0& 0\\ 0& U_5& V_1& 0& 0& 0& 0& 0& 0& 0\\ 0& 2U_5& 0& 0& 0& 0& 0& 0& 0& 0\\ 2U_4& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 2U_5& 0& 0& 0& 0& 0& 0& 0& 0\end{array}\right),\hfill \\ \hfill & Q_8=\left(\begin{array}{cccccccccc}{F}_{33}& F_{33}& 0& G_3& 0& 0& 0& 0& 0& 0\\ 0& F_{33}& F_{33}& 0& G_3& 0& 0& 0& 0& 0\\ F_{22}& 0& 0& 0& 0& G_2& 0& 0& 0& 0\\ F_{11}& 0& 0& 0& 0& 0& G_1& 0& 0& 0\\ 0& F_{22}& 0& 0& 0& 0& 0& G_2& 0& 0\\ 0& F_{11}& 0& 0& 0& 0& 0& 0& G_1& 0\\ 0& 0& F_{22}& 0& 0& 0& 0& 0& 0& G_2\end{array}\right),\hfill \end{array}\hfill \end{array}$$

(47)

$$\begin{array}{cc}\hfill & \begin{array}{cc}\hfill & P_{10}=\left(\begin{array}{cccccccc}0& 0& E_{22}& 0& 0& E_{11}& 0& 0\\ 2E_{22}& 0& 0& 0& 0& 0& 0& E_{11}\\ 0& 0& 0& E_{22}& 0& 0& E_{11}& 0\\ 0& E_{22}& 0& 0& E_{11}& 0& 0& 0\\ E_2& 0& 0& 0& 0& 0& 0& 0\\ 0& E_2& 0& 0& 0& 0& 0& 0\\ 0& 0& E_2& 0& 0& 0& 0& 0\\ 0& 0& 0& E_2& 0& 0& 0& 0\\ 0& 0& 0& 0& E_1& 0& 0& 0\\ 0& 0& 0& 0& 0& E_1& 0& 0\\ 0& 0& 0& 0& 0& 0& E_1& 0\\ 0& 0& 0& 0& 0& 0& 0& E_1\end{array}\right),P_9=\left(\begin{array}{ccc}0& E_{11}& 0\\ E_{22}& 0& E_{11}\\ E_2& 0& 0\\ 0& E_1& 0\\ 0& 0& E_1\end{array}\right),\hfill \end{array}\hfill \end{array}$$

(48)

$$\begin{array}{cc}\hfill & \begin{array}{cc}\hfill & Q_9=\left(\begin{array}{cccccc}{F}_{33}& F_{33}& G_3& 0& 0& 0\\ F_{22}& 0& 0& G_2& 0& 0\\ F_{11}& 0& 0& 0& G_1& 0\\ 0& F_{22}& 0& 0& 0& G_2\end{array}\right),S_9=\left(\begin{array}{cccccc}2T& 2T& 2V_2& 2V_3& 2V_4& 2V_3\\ 0& T& 0& 0& 0& W_4\\ 0& U_4& 0& 0& 0& 0\\ 2U_5& 2U_1& 0& 0& 0& 0\\ 0& U_5& 0& 0& 0& 0\end{array}\right),\hfill \end{array}\hfill \end{array}$$

$$\begin{array}{cc}\hfill & \begin{array}{cc}\hfill & Q_{10}=\left(\begin{array}{cccccccccccc}{F}_{33}& F_{33}& F_{33}& 0& G_3& 0& 0& 0& 0& 0& 0& 0\\ 0& F_{33}& F_{33}& F_{33}& 0& G_3& 0& 0& 0& 0& 0& 0\\ F_{22}& 0& 0& 0& 0& 0& G_2& 0& 0& 0& 0& 0\\ F_{11}& 0& 0& 0& 0& 0& 0& G_1& 0& 0& 0& 0\\ 0& F_{22}& 0& 0& 0& 0& 0& 0& G_2& 0& 0& 0\\ 0& 0& F_{11}& 0& 0& 0& 0& 0& 0& G_1& 0& 0\\ 0& 0& F_{22}& 0& 0& 0& 0& 0& 0& 0& G_2& 0\\ 0& 0& 0& F_{22}& 0& 0& 0& 0& 0& 0& 0& G_2\end{array}\right),\hfill \end{array}\hfill \end{array}$$

(49)

$$\begin{array}{cc}\hfill & \begin{array}{cc}\hfill & S_{10}=\left(\begin{array}{ccccccccccccc}2T& 0& 0& 0& 0& 0& 0& 2V_3& 2V_4& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& T& 0& 0& 0& 0& 0& 0& 0& 0& V_3\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& U_4& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 2U_4& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& W_1& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 2U_5& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\end{array}\right),\hfill \end{array}\hfill \end{array}$$

(50)

and

$$\begin{array}{cc}\hfill & W_1=F_1{F}_{11}+F_2{F}_{22},W_2=F_3{F}_{22}+F_4{F}_{33},\hfill \\ \hfill & W_3=E_{22}{H}_2+E_{22}{H}_3,W_4=E_{11}{H}_2+E_{22}{H}_3,\hfill \end{array}$$

(51)

$$\begin{array}{cc}\hfill & V_1=E_{11}{H}_2,V_3=E_{22}{H}_3,V_2=E_{22}{H}_4,V_4=E_{11}{H}_1,\hfill \end{array}$$

(52)

$$\begin{array}{cc}\hfill & U_1=F_1{F}_{11},U_2=F_2{F}_{11},U_3=F_3{F}_{22},U_4=F_4{F}_{33},U_5=F_2{F}_{22}.\hfill \end{array}$$

(53)

In this case, the general solution to the system of the matrix Equation (4) can be expressed as follows:

$$\begin{array}{cc}\hfill & X_1=E_1^{-}{F}_1+L_{E_1}{H}_1{G}_1^{-}+L_{E_1}{U}_1{R}_{G_1},\hfill \\ \hfill & X_2=E_1^{-}{F}_2+L_{E_1}{H}_2{G}_2^{-}+L_{E_1}{U}_2{R}_{G_2},\hfill \\ \hfill & X_3=E_2^{-}{F}_3+L_{E_2}{H}_3{G}_2^{-}+L_{E_2}{U}_3{R}_{G_2},\hfill \\ \hfill & X_4=E_2^{-}{F}_4+L_{E_2}{H}_4{G}_3^{-}+L_{E_2}{U}_4{R}_{G_3},\hfill \end{array}$$

where

$$\begin{array}{cc}\hfill & U_1=A_1^{-}\left[(C_c-A_2{X}_4{B}_3)L_{B_2}+R_{A_1}(C_c-A_2{X}_4{B}_3)\right]B_4^{-}+L_{A_1}{V}_1+V_2{R}_{B_4},\hfill \\ \hfill & U_2=A_1^{-}(C_c-A_1{X}_1{B}_1-A_2{X}_3{B}_2-A_2{X}_4{B}_3)B_2^{-}+V_3{R}_{B_2}-L_{A_1}{V}_4,\hfill \\ \hfill & U_3=M^{-}\left[(C_c-A_2{X}_4{B}_3)L_{B_2}+R_{A_1}(C_c-A_2{X}_4{B}_3)\right]B_2^{-}+L_M{V}_5+V_6{R}_{B_2},\hfill \\ \hfill & U_4=\varphi +L_{A_{11}}{L}_{M_1}{V}_7+V_8{R}_{N_1}{R}_{B_{11}}+L_{A_{11}}{V}_9{R}_{B_{22}}+L_{A_{22}}{V}_{10}{R}_{B_{11}},\hfill \\ \hfill & or\hfill \\ \hfill & U_4=A_{33}^{-}{C}_{33}{B}_{33}^{-}-L_{A_{33}}{V}_{11}-V_{12}{R}_{B_{33}},\hfill \\ \hfill & V_7=S_1\left[M_2^{-}\left(E_1-L_{A_{11}}{U}_3{R}_{B_{22}}-L_{A_{22}}{U}_4{R}_{B_{11}}\right)-M_2^{-}{T}_7{N}_2+L_{M_2}{T}_6\right],\hfill \\ \hfill & V_8=\left[R_{M_2}\left(E_1-L_{A_{11}}{U}_3{R}_{B_{22}}-L_{A_2}{U}_4{R}_{B_{11}}\right)N_2^{-}+M_2{M}_2^{-}{T}_7+T_8{R}_{N_2}\right]S_2,\hfill \\ \hfill & V_9=A^{-}E{B}^{-}-A^{-}C{M}^{-}{R}_{A}E{B}^{-}-A^{-}S{C}^{-}E{L}_B{N}^{-}D{B}^{-}-A^{-}S{T}_3{R}_{N}D{B}^{-}\hfill \\ \hfill & +L_A{T}_1+T_2{R}_B,\hfill \\ \hfill & V_{10}=M^{-}{R}_{A}E{D}^{-}+L_M{S}^{-}S{C}^{-}E{L}_B{N}^{-}+L_M{L}_S{T}_4+L_M{T}_3{R}_N+T_5{R}_D,\hfill \\ \hfill & V_{11}=S_3\left[M_2^{-}\left(E_1-L_{A_{11}}{U}_3{R}_{B_{22}}-L_{A_{22}}{U}_4{R}_{B_{11}}\right)-M_2^{-}{T}_7{N}_2+L_{M_2}{T}_6\right],\hfill \\ \hfill & V_{12}=\left[R_{M_2}\left(E_1-L_{A_{11}}{U}_3{R}_{B_{22}}-L_{A_{22}}{U}_4{R}_{B_{11}}\right)N_2^{-}+M_2{M}_2^{-}{T}_7+T_8{R}_{N_2}\right]S_4,\hfill \end{array}$$

$V_i(i=\overline{1,6})$ are arbitrary matrices over $\mathbb{H}$ with appropriate sizes, and $T_i(i=\overline{1,8})$ are arbitrary matrices over $\mathbb{H}$ with appropriate sizes.

Proof. 

$\left(1\right)\iff \left(2\right)$ Clearly, the system of matrix Equation (4) is solvable if—and only if—both of the following are consistent:

$$\begin{array}{cc}\hfill & E_1{X}_1=F_1,X_1{G}_1=H_1,\hfill \\ \hfill & E_1{X}_2=F_2,X_2{G}_2=H_2,\hfill \\ \hfill & E_2{X}_3=F_3,X_3{G}_2=H_3,\hfill \\ \hfill & E_2{X}_4=F_4,X_4{G}_3=H_4\hfill \end{array}$$

(54)

and

$$\begin{array}{cc}\hfill E_{11}{X}_1{F}_{11}+& E_{11}{X}_2{F}_{22}+E_{22}{X}_3{F}_{22}+E_{22}{X}_4{F}_{33}=T.\hfill \end{array}$$

(55)

It follows from Lemma 4 that the system (54) has a solution if—and only if— (35) holds, and

$$\begin{array}{cc}\hfill & R_{E_1}{F}_1=0,R_{E_1}{F}_2=0,R_{E_2}{F}_3=0,R_{E_2}{F}_4=0,\hfill \\ \hfill & H_1{L}_{G_1}=0,H_2{L}_{G_2}=0,H_3{L}_{G_2}=0,H_4{L}_{G_3}=0.\hfill \end{array}$$

(56)

Under these conditions, the expression of general solution to (54) can be expressed as follows:

$$\begin{array}{cc}\hfill & X_1=E_1^{-}{F}_1+L_{E_1}{H}_1{G}_1^{-}+L_{E_1}{U}_1{R}_{G_1},\hfill \\ \hfill & X_2=E_1^{-}{F}_2+L_{E_1}{H}_2{G}_2^{-}+L_{E_1}{U}_2{R}_{G_2},\hfill \\ \hfill & X_3=E_2^{-}{F}_3+L_{E_2}{H}_3{G}_2^{-}+L_{E_2}{U}_3{R}_{G_2},\hfill \\ \hfill & X_4=E_2^{-}{F}_4+L_{E_2}{H}_4{G}_3^{-}+L_{E_2}{U}_4{R}_{G_3},\hfill \end{array}$$

(57)

where $U_i(i=\overline{1,4})$ are arbitrary matrices over $\mathbb{H}$ with appropriate sizes.

Next, substituting (57) into (55) yields the following:

$$\begin{array}{c}\hfill A_1{U}_1{B}_1+A_1{U}_2{B}_2+A_2{U}_3{B}_2+A_2{U}_4{B}_3=C_c,\end{array}$$

(58)

where $A_i$, $B_j$, $(i=\overline{1,2},j=\overline{1,3})$ are defined as (34). By Theorem 1, the matrix Equation (58) is solvable if—and only if—(37) holds. In this case, the general solution to matrix Equation (58) can be expressed as follows:

$$\begin{array}{cc}\hfill & U_1=A_1^{-}\left[(C_c-A_2{X}_4{B}_3)L_{B_2}+R_{A_1}(C_c-A_2{X}_4{B}_3)\right]B_4+L_{A_1}{V}_1+V_2{R}_{B_4},\hfill \\ \hfill & U_2=A_1^{-}(C_c-A_1{X}_1{B}_1-A_2{X}_3{B}_2-A_2{X}_4{B}_3)B_2+V_3{R}_{B_2}-L_{A_1}{V}_4,\hfill \\ \hfill & U_3=M^{-}\left[(C_c-A_2{X}_4{B}_3)L_{B_2}+R_{A_1}(C_c-A_2{X}_4{B}_3)\right]B_2+L_M{V}_5+V_6{R}_{B_2},\hfill \\ \hfill & U_4=\varphi +L_{A_{11}}{L}_{M_1}{V}_7+V_8{R}_{N_1}{R}_{B_{11}}+L_{A_{11}}{V}_9{R}_{B_{22}}+L_{A_{22}}{V}_{10}{R}_{B_{11}},\hfill \\ \hfill & or{U}_4=A_{33}^{-}{C}_{33}{B}_{33}^{-}-L_{A_{33}}{V}_{11}-V_{12}{R}_{B_{33}},\hfill \\ \hfill & V_7=S_1\left[M_2^{-}\left(E_1-L_{A_{11}}{U}_3{R}_{B_{22}}-L_{A_{22}}{U}_4{R}_{B_{11}}\right)-M_2^{-}{T}_7{N}_2+L_{M_2}{T}_6\right],\hfill \\ \hfill & V_8=\left[R_{M_2}\left(E_1-L_{A_{11}}{U}_3{R}_{B_{22}}-L_{A_2}{U}_4{R}_{B_{11}}\right)N_2^{-}+M_2{M}_2^{-}{T}_7+T_8{R}_{N_2}\right]S_2,\hfill \\ \hfill & V_9=A^{-}E{B}^{-}-A^{-}C{M}^{-}{R}_{A}E{B}^{-}-A^{-}S{C}^{-}E{L}_B{N}^{-}D{B}^{-}-A^{-}S{T}_3{R}_{N}D{B}^{-}\hfill \\ \hfill & +L_A{T}_1+T_2{R}_B,\hfill \\ \hfill & V_{10}=M^{-}{R}_{A}E{D}^{-}+L_M{S}^{-}S{C}^{-}E{L}_B{N}^{-}+L_M{L}_S{T}_4+L_M{T}_3{R}_N+T_5{R}_D,\hfill \\ \hfill & V_{11}=S_3\left[M_2^{-}\left(E_1-L_{A_{11}}{U}_3{R}_{B_{22}}-L_{A_{22}}{U}_4{R}_{B_{11}}\right)-M_2^{-}{T}_7{N}_2+L_{M_2}{T}_6\right],\hfill \\ \hfill & V_{12}=\left[R_{M_2}\left(E_1-L_{A_{11}}{U}_3{R}_{B_{22}}-L_{A_{22}}{U}_4{R}_{B_{11}}\right)N_2^{-}+M_2{M}_2^{-}{T}_7+T_8{R}_{N_2}\right]S_4,\hfill \end{array}$$

where $V_i(i=\overline{1,6})$ are arbitrary matrices over $\mathbb{H}$ with appropriate sizes, $A_{ii},B_{ii},C_{ii}(i=\overline{1,3}),M_1,M_2,N_1,N_2$, $A,B,D,E,S,M$ and N are defined as (34), $T_i(i=\overline{1,8})$ are arbitrary matrices over $\mathbb{H}$, with appropriate sizes.

Hence, the system of matrix Equation (54) and the matrix Equation (55) are consistent if—and only if—(35), (56), and (37) hold,

Now, we show that (56) $\iff$ (36). It follows from Lemma 2 that the following is true:

$$\begin{array}{cc}\hfill & R_{E_1}{F}_1=0\iff r(F_1,E_1)=r\left(E_1\right),R_{E_1}{F}_2=0\iff r(F_2,E_1)=r\left(E_1\right),\hfill \\ \hfill & R_{E_2}{F}_3=0\iff r(F_3,E_2)=r\left(E_2\right),R_{E_2}{F}_4=0\iff r(F_4,E_2)=r\left(E_2\right),\hfill \\ \hfill & H_1{L}_{G_1}=0\iff r\left(\begin{array}{c}{H}_1\\ G_1\end{array}\right)=r\left(G_1\right),H_2{L}_{G_2}=0\iff r\left(\begin{array}{c}{H}_2\\ G_2\end{array}\right)=r\left(G_2\right),\hfill \\ \hfill & H_3{L}_{G_2}=0\iff r\left(\begin{array}{c}{H}_3\\ G_2\end{array}\right)=r\left(G_2\right),H_4{L}_{G_3}=0\iff r\left(\begin{array}{c}{H}_4\\ G_3\end{array}\right)=r\left(G_3\right).\hfill \end{array}$$

(59)

According to (59), we obtain (56) $\iff$ (36). To sum up, the system of matrix Equation (54) and the matrix Equation (55) are consistent if—and only if—(35)–(37) hold.

$\left(2\right)\iff \left(3\right)$ In view of (59), we obtain (36)$\iff$ (38).

Next, we turn our attention to show that (37) holds if—and only if—(39) holds. According to Theorem 1, we can find that (37) holds if—and only if—(10)–(19) hold. Hence, we need prove $(9+i)\iff$ (39)$(i=\overline{1,10})$ when we show that (37) holds if—and only if—(39) holds. We need to use the following fact to prove $(9+i)\iff$ (39)$(i=\overline{1,10})$:

It is easy to know that there exists a solution, according to $X_1^0,X_2^0,X_3^0,X_4^0$, such that the following is true:

$$\begin{array}{cc}\hfill & E_1{X}_1^0=F_1,X_1^0{G}_1=H_1,\hfill \\ \hfill & E_1{X}_2^0=F_2,X_2^0{G}_2=H_2,\hfill \\ \hfill & E_2{X}_3^0=F_3,X_3^0{G}_2=H_3,\hfill \\ \hfill & E_2{X}_4^0=F_4,X_4^0{G}_3=H_4,\hfill \\ \hfill E_{11}{X}_1^0{F}_{11}+& E_{11}{X}_2^0{F}_{22}+E_{22}{X}_3^0{F}_{22}+E_{22}{X}_4^0{F}_{33}=T,\hfill \end{array}$$

(60)

where

$$\begin{array}{cc}\hfill & X_1^0=E_1^{-}{F}_1+L_{E_1}{H}_1{G}_1^{-},X_2^0=E_1^{-}{F}_2+L_{E_1}{H}_2{G}_2^{-},\hfill \\ \hfill & X_3^0=E_2^{-}{F}_3+L_{E_2}{H}_3{G}_2^{-},X_4^0=E_2^{-}{F}_4+L_{E_2}{H}_4{G}_3^{-}.\hfill \end{array}$$

Let $T_0=T-(E_{11}{X}_1^0{F}_{11}+E_{11}{X}_2^0{F}_{22}+E_{22}{X}_3^0{F}_{22}+E_{22}{X}_4^0{F}_{33})$. We first show the following: $(9+i)\iff$ (39) for $i=1$, $(9+i)\iff$ (39) for $i=2$, $(9+i)\iff$ (39) for $i=3$, $(9+i)\iff$ (39) for $i=4$, and $(9+i)\iff$ (39) for $i=5$.

In fact, when $i=1$, by Lemma 2, (60) and elementary operations, we obtain the following:

$$\begin{array}{cc}\hfill & (9+i)\iff r\left(\begin{array}{ccc}2T_0& E_{22}{L}_{E_2}& E_{11}{L}_{E_1}\\ R_{G_1}{F}_{11}& 0& 0\\ R_{G_2}{F}_{22}& 0& 0\end{array}\right)=r\left(\begin{array}{cc}{E}_{22}{L}_{E_2},& E_{11}{L}_{E_1}\end{array}\right)+r\left(\begin{array}{c}{R}_{G_1}{F}_{11}\\ R_{G_2}{F}_{22}\end{array}\right)\iff \hfill \\ \hfill & r\left(\begin{array}{ccccc}2T_0& E_{22}& E_{11}& 0& 0\\ F_{11}& 0& 0& G_1& 0\\ F_{22}& 0& 0& 0& G_2\\ 0& E_2& 0& 0& 0\\ 0& 0& E_{11}& 0& 0\end{array}\right)=r\left(\begin{array}{cc}{E}_{22}& E_{11}\\ E_2& 0\\ 0& E_{11}\end{array}\right)+r\left(\begin{array}{ccc}{F}_{11}& G_1& 0\\ F_{22}& 0& G_2\end{array}\right)\iff \left(39\right).\hfill \end{array}$$

Similarly, we can show that $(9+i)\iff$ (39) for $i=2$, $(9+i)\iff$ (39) for $i=3$, $(9+i)\iff$ (39) for $i=4$ and $(9+i)\iff$ (39) for $i=5$, where $P_i,Q_i,S_i$ and $O_i(i=\overline{1,5})$ in (39) are defined as (40), (41), (42), (43), and (44), respectively, $W_i(i=\overline{1,3})$ are defined as (51), $U_j(j=\overline{1,5})$ are defined as (53), and $V_k(k=\overline{1,4})$ are defined as (52).

Second, we show that $(9+i)\iff$ (39) for $i=6$, $(9+i)\iff$ (39) for $i=7$, $(9+i)\iff$ (39) for $i=8$, $(9+i)\iff$ (39) for $i=9$ and $(9+i)\iff$ (39) for $i=10$. In fact, when $i=6$, it follows from Lemma 2, (60), and elementary operations, that the following is true:

$$\begin{array}{cc}\hfill & (9+i)\iff r\left(\begin{array}{cccccc}2T_0& 0& 0& E_{22}{L}_{E_2}& E_{11}{L}_{E_1}& 0\\ R_{G_3}{F}_{33}& 0& R_{G_3}{F}_{33}& 0& 0& 0\\ 0& E_{22}{L}_{E_2}& 2T_0& 0& 0& E_{11}{L}_{E_1}\\ R_{G_1}{F}_{11}& 0& 0& 0& 0& 0\\ 0& 0& R_{G_1}{F}_{11}& 0& 0& 0\\ 0& 0& R_{G_2}{F}_{22}& 0& 0& 0\\ R_{G_2}{F}_{22}& 0& 0& 0& 0& 0\end{array}\right)\hfill \\ \hfill & =r\left(\begin{array}{cccc}{E}_{22}{L}_{E_2}& E_{11}{L}_{E_1}& 0& 0\\ 0& 0& E_{22}{L}_{E_2}& E_{11}{L}_{E_1}\end{array}\right)+r\left(\begin{array}{cc}{R}_{G_3}{F}_{33}& R_{G_3}{F}_{33}\\ R_{G_2}{F}_{22}& 0\\ R_{G_1}{F}_{11}& 0\\ 0& R_{G_2}{F}_{22}\\ 0& R_{G_1}{F}_{11}\end{array}\right)\hfill \\ \hfill & \iff r\left(\begin{array}{ccccccccccc}2T_0& 0& 0& E_{22}& E_{11}& 0& 0& 0& 0& 0& 0\\ F_{33}& 0& F_{33}& 0& 0& 0& G_3& 0& 0& 0& 0\\ 0& E_{22}& 2T_0& 0& 0& E_{11}& 0& 0& 0& 0& 0\\ F_{11}& 0& 0& 0& 0& 0& 0& G_1& 0& 0& 0\\ 0& 0& F_{11}& 0& 0& 0& 0& 0& G_1& 0& 0\\ 0& 0& F_{22}& 0& 0& 0& 0& 0& 0& G_2& 0\\ F_{22}& 0& 0& 0& 0& 0& 0& 0& 0& 0& G_2\\ 0& E_2& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& E_2& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& E_1& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& E_1& 0& 0& 0& 0& 0\end{array}\right)\hfill \\ \hfill & =r\left(\begin{array}{cccc}{E}_{22}& E_{11}& 0& 0\\ 0& 0& E_{22}& E_{11}\\ E_2& 0& 0& 0\\ 0& E_1& 0& 0\\ 0& 0& E_2& 0\\ 0& 0& 0& E_1\end{array}\right)+r\left(\begin{array}{ccccccc}{F}_{33}& F_{33}& G_3& 0& 0& 0& 0\\ F_{22}& 0& 0& G_2& 0& 0& 0\\ F_{11}& 0& 0& 0& G_1& 0& 0\\ 0& F_{22}& 0& 0& 0& G_2& 0\\ 0& F_{11}& 0& 0& 0& 0& G_1\end{array}\right)\iff \left(39\right).\hfill \end{array}$$

Similarly, we can show that $(9+i)\iff \left(\right)$ (39) for $i=7$, $(9+i)\iff \left(\right)$ (39) for $i=8$, $(9+i)\iff \left(\right)$ (39) for $i=9$ and $(9+i)\iff \left(\right)$ (39) for $i=10$, where $P_i,Q_i,S_i(i=6,7,8)$, and in (39), are defined as (45), (46), (47), respectively, $P_i,Q_i$ and $S_i(i=9,10)$ in (39), are defined as (48), (49), and (50). $W_i(i=\overline{1,3})$ are defined as (51), $U_j(j=\overline{1,5})$ are defined as (53), and $V_k(k=\overline{1,4})$ are defined as (52). The proof is completed. □

4. Conclusions

We have established some necessary and sufficient conditions for the existence of the solution to quaternion matrix Equation (1), and derived a formula of its general solution when it is solvable. As an application of (1), we have investigated some necessary and sufficient conditions for the system of matrix Equation (4) to be consistent, as well as the expression of its general solution, and presented a numerical example to emphasize our main results.