Analysis of Solvability and Representation of General Solutions for Anti-Hermitian Constrained Quaternion Matrix Equations

1. Introduction

We consider the set of matrices with entries that belong to the quaternion skew field determined by $\mathbb{H}=\{t_0+t_1\mathbf{i}+t_2\mathbf{j}+t_3\mathbf{k}\mid {\mathbf{i}}^2={\mathbf{j}}^2={\mathbf{k}}^2=\mathbf{ijk}=-1,t_0,t_1,t_2,t_3\in \mathbb{R}\}$. For an arbitrary quaternion $t=t_0+t_1\mathbf{i}+t_2\mathbf{j}+t_3\mathbf{k}\in \mathbb{H}$, its conjugate quaternion is defined by $\overline{t}=t_0-t_1\mathbf{i}-t_2\mathbf{j}-t_3\mathbf{k}$. $\mathbb{C}$ and $\mathbb{R}$ represent the complex and real number field, respectively. For any $A\in {\mathbb{H}}^{m\times n}$, the matrix $A^{*}\in {\mathbb{H}}^{n\times m}$ indicates the conjugate transpose of matrix A. A matrix $A\in {\mathbb{H}}^{n\times n}$ is said to be Hermitian when $A^{*}=A$, and it is anti-Hermitian when $A^{*}=A$. Sometimes, instead of referring to "anti-Hermitian," the term "skew-Hermitian" is used, especially when describing quaternion matrices. Note that for any $A\in {\mathbb{H}}^{m\times n}$, the sum of it and its conjugate transpose $\check{A}:={\displaystyle \frac{1}{2}}(A+A^{*})$ is Hermitian, and their difference

$$\begin{array}{c}\hfill \widehat{A}:={\displaystyle \frac{1}{2}}(A-A^{*})=-{\widehat{A}}^{*},\end{array}$$

(1)

yields its corresponding skew-Hermitian matrix, and $A=\check{A}+\widehat{A}$ holds.

Both symbols $r\left(A\right)$ and $\mathrm{rank}A$ denote the rank of A. I denotes an identity matrix with suitable shape. The Moore-Penrose inverse (MPI) of $A\in {\mathbb{H}}^{m\times n}$ is denoted by $W:=A^{†}\in {\mathbb{H}}^{n\times m}$ and is defined by the four Penrose’s equations

$$\begin{array}{cc}\hfill & \left(1\right)AWA=A,\left(2\right)WAW=W,\left(3\right){\left(AW\right)}^{*}=AW,\left(4\right){\left(WA\right)}^{*}=WA.\hfill \end{array}$$

Furthermore, $L_A=I-A^{†}A$ and $R_A=I-A{A}^{†}$ are pair of projectors produced by A, and

$$\begin{array}{c}\hfill L_A={\left(L_A\right)}^{*}={\left(L_A\right)}^2=L_A^{†},R_A={\left(R_A\right)}^2={\left(R_A\right)}^{*}=R_A^{†},L_{A^{*}}=R_A,R_{A^{*}}=L_A.\end{array}$$

The introduction of quaternionic mathematics has significantly expanded the scope of applied mathematical fields [1,2,3,4]. Quaternions are fundamental in the description of three-dimensional rotations and find widespread application in computer graphics, robotics, navigation, quantum physics, mechanics, and signal processing, as extensively documented in [5]. Recent advancements in the development of explicit formulas and determinantal representations for quaternion matrix equations have enabled novel approaches to solving both classical and quaternion systems [6,7,8,9,10]. The increasing interest among scientists in the application of quaternions across diverse practical domains has spurred the investigation of anti-Hermitian solutions for quaternion systems of matrix equations.

Numerous problems in various engineering disciplines, including linear descriptor systems, system design, singular system control [11,12,13], perturbation theory [14], and feedback control [15], necessitate the solution of Sylvester-type matrix equations. For example, Bai [16] examined the iterative solution of $A_{1}X+X{A}_2=B$. Roth [17] established the consistency condition for the existence of solutions to $A_{1}X+Y{A}_2=B$, and researchers in [18] derived its general solution.

Recently, the general solution of

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

(2)

was explored in [19]. In [20], some solvability conditions to (2)are also researched. The condition number of (2) was also evaluated in [21]. The constraint solution to (2) are explored by Wang et al. [22]. Some practical necessary and sufficient conditions for

$$\begin{array}{c}\hfill A_1{X}_1+Z_1{B}_1=C_1,A_2{Z}_1+X_2{B}_2=C_2,\end{array}$$

to have a general solution are presented by Wang and He in [23].

The generalized Sylvester matrix equation

$$AXB+C\mathbf{Y}D=E,$$

(3)

has been extensively studied. Baksalari and Kala [24] provided a comprehensive solution to the complex equation (3) employing the Moore-Penrose inverse. This result later extended and developed to quaternion equations by Wang [25,26].

Liu evaluated the Hermitian solutions to Sylvester-type equation

$$\begin{array}{c}\hfill A_{1}X{A}_1^{*}+B_{1}Y{B}_1^{*}=C_1\end{array}$$

(4)

over $\mathbb{C}$ in [27]. Additionally, [28] investigated equation (4) in its findings. A nonlinear Hermitian expression was also explored in [31]. Kyrchei, in [29,30], utilized the established theory of row-column noncommutative determinants to provide Cramer’s rules for the solution to equation (3) and Hermitian solution of equation (4) within the quaternion skew field $\mathbb{H}$. More recently, Hajarian, in [32], developed an algorithm for determining the solution to the system of the generalized Sylvester matrix equations

$$\begin{array}{c}\hfill A_{1}X{B}_1+C_{1}Y{D}_1=E_1,\\ \hfill A_{2}Z{B}_2+C_{2}Y{D}_2=E_2.\end{array}$$

Some latest development to the general solution of Sylvester-type matrix equations can be found in [33,34,35,36,37,38,39,40]. Some iterative algorithms of solving coupled matrix equations can be found in [41,42].

Very recently, researchers in [43] examined the general solution of

$$\begin{array}{cc}\hfill & A_{1}U{A}_1^{*}+B_{1}V{B}_1^{*}=C_1,C_1=-C_1^{*}\hfill \\ \hfill & A_{2}W{A}_2^{*}+B_{2}V{B}_2^{*}=C_2,C_2=-C_2^{*},\hfill \end{array}$$

(5)

with some practical necessary and sufficient conditions when this system is consistent.

Driven by the aforementioned research and the substantial applicability of generalized Sylvester matrix equations across diverse applied fields, this paper investigates the constrained anti-Hermitian Sylvester matrix equations:

$$\begin{array}{cc}\hfill & A_{3}X=C_3,X{B}_3=C_4,X^{*}=-X,\hfill \\ \hfill & A_{4}Y=C_5,Y{B}_4=C_6,Y^{*}=-Y,\hfill \\ \hfill & A_{5}Z=C_7,Z{B}_5=C_8,Z^{*}=-Z,\hfill \\ \hfill & A_{1}X{A}_1^{*}+B_{1}Y{B}_1^{*}=C_1,C_1=-C_1^{*}\hfill \\ \hfill & A_{2}Z{A}_2^{*}+B_{2}Y{B}_2^{*}=C_2,C_2=-C_2^{*},\hfill \end{array}$$

(6)

over the quaternion skew field $\mathbb{H}$. We will establish the anti-Hermitian solution to the system (6) using the Moore-Penrose inverse and provide an algorithm for its determination, employing determinantal representations of the Moore-Penrose inverse as developed in [44]. The results pertaining to system (6) will significantly broaden the application of anti-Hermitian Sylvester matrix equations in various domains. The primary objective of this paper is to compute the general solution of (6) when it is solvable. The general solution of:

$$\begin{array}{c}\hfill A_{4}X-{\left(A_{4}X\right)}^{*}+B_{4}Y{B}_4^{*}+C_{4}Z{C}_4^{*}=D_4,D_4=-D_4^{*},Y=-Y^{*},Z=-Z^{*},\end{array}$$

(7)

plays a fundamental role in deriving the main findings of this paper over $\mathbb{H}$ with anti-Hermitian properties.

Recent research has highlighted the growing interest in Sylvester-type matrix equations for dual quaternions, which extend Hamilton quaternions (see, e.g., [45]). Notably, the synergistic relationship between different quaternion algebras facilitates valuable insights, especially in the development of techniques for solving matrix equations.

This paper is structured into four sections. Section 2 establishes the general solution of (6), including a special case. Section 3 presents an algorithm and a numerical example for the anti-Hermitian solution of (6). Finally, Section 4 provides a conclusion to this research.

2. General solution and solvability condition of (6)

Some significant results are given at the outset of this section which have eminent function in constructing the main result of this paper.

Lemma 1

([46]). Let $K\in {\mathbb{C}}^{m\times n},P\in {\mathbb{C}}^{m\times t},Q\in {\mathbb{C}}^{l\times n}$. Then

$$\begin{array}{cc}\hfill r\left[\begin{array}{c}K\\ Q\end{array}\right]& -r\left(K\right)=r\left(Q{L}_K\right),r\left[\begin{array}{cc}K& P\end{array}\right]-r\left(P\right)=r\left(R_{P}K\right),\hfill \\ \hfill & r\left[\begin{array}{cc}K& P\\ Q& 0\end{array}\right]-r\left(P\right)-r\left(Q\right)=r\left(R_{P}K{L}_Q\right).\hfill \end{array}$$

Lemma 2.

[47]. Let $A_2,B_2,C_2$ and $D_2$ be given with conformable dimensions over $\mathbb{H}$. Set

$$E=\left[\begin{array}{c}{A}_2\\ -B_2^{*}\end{array}\right],F=\left[\begin{array}{c}{C}_2\\ D_2^{*}\end{array}\right].$$

Then the system $A_{2}Y=C_2,Y{B}_2=D_2$ has the skew-Hermitian solution if and only if $R_{E}F=0$ and $E{F}^{*}=-F{E}^{*}.$ Under these terms, its general skew-Hermitian solution is

$$Y=E^{†}F-{\left(E^{†}F\right)}^{*}+E^{†}E{\left(E^{†}F\right)}^{*}+L_{E}V{L}_E^{*},$$

where $V=-V^{*}$ is a free matrix over $\mathbb{H}$ with conformable size.

Lemma 3.

[6]. Let $A_4,B_4,C_4$ and $D_4=-D_4^{*},$ be known coefficient matrices in (7) over $\mathbb{C}$ with agreeable sizes. Assume

$$A=R_{A_4}{B}_4,B=R_{A_4}{C}_4,C=R_{A_4}{D}_4{R}_{A_4},M=R_{A}B,S=B{L}_M.$$

Then the conditions given below are alike:

(1)

The system (7) has a solution $(X,Y,Z)$, where Y and Z are anti-Hermitian matrices.

(2)

The coefficient matrices in (7) satisfy:

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

(3)

$$\begin{array}{ccc}& & r\left[\begin{array}{cccc}{D}_4& A_4& B_4& C_4\\ A_4^{*}& 0& 0& 0\end{array}\right]=r\left[\begin{array}{ccc}{A}_4& B_4& C_4\end{array}\right]+r\left(A_4\right),\hfill \\ & & r\left[\begin{array}{ccc}{D}_4& A_4& B_4\\ A_4^{*}& 0& 0\\ C_4^{*}& 0& 0\end{array}\right]=r\left[\begin{array}{cc}{A}_4& B_4\end{array}\right]+r\left[\begin{array}{cc}{A}_4& C_4\end{array}\right].\hfill \end{array}$$

Under these conditions, $X,Y^{*}=-Y$ and $Z^{*}=-Z$ are given below

$$\begin{array}{cc}\hfill X=& A_4^{†}[D_4-B_{4}Y{B}_4^{*}-C_{4}Z{C}_4^{*}]-{\displaystyle \frac{1}{2}}{A}_4^{†}[D_4-B_{4}Y{B}_4^{*}-C_{4}Z{C}_4^{*}]{\left(A_4^{†}\right)}^{*}{A}_4^{*}-U_2^{*}{\left(A_4^{†}\right)}^{*}{A}_4^{*}-\hfill \\ & -A_4^{†}{U}_2{A}_4^{*}+L_{A_4}{U}_1,\hfill \\ \hfill Y=& -Y^{*}=A^{†}C{\left(A^{†}\right)}^{*}-{\displaystyle \frac{1}{2}}(A^{†}B{M}^{†}C[I+{\left(B^{†}\right)}^{*}{S}^{*}]{\left(A^{†}\right)}^{*}-A^{†}[I+S{B}^{†}]C{\left(M^{†}\right)}^{*}{B}^{*}{\left(A^{†}\right)}^{*})-\hfill \\ & -A^{†}S{U}_6{S}^{*}{\left(A^{†}\right)}^{*}+L_A{U}_4-U_4^{*}{L}_A,\hfill \\ \hfill Z=& -Z^{*}={\displaystyle \frac{1}{2}}{M}^{†}C{\left(B^{†}\right)}^{*}[I+{\left(S^{†}S\right)}^{*}]-{\displaystyle \frac{1}{2}}(I+S^{†}S)B^{†}C{\left(M^{†}\right)}^{*}+L_M{U}_6{L}_M+\hfill \\ \hfill & +U_5{L}_B-L_B{U}_5^{*}+L_M{L}_S{U}_3-{\left(L_M{L}_S{U}_3\right)}^{*},\hfill \end{array}$$

where $U_1,\cdots ,U_5$ and $U_6=-U_6^{*}$ are free matrices with acceptable dimensions.

Lemma 3 has solid role in obtaining the main theorem of this paper. Now we present the main result of this paper.

Theorem 1.

Let $A_i\in {\mathbb{H}}^{m\times n}$, $B_i\in {\mathbb{H}}^{m\times q}$ for all $i=1,\dots ,5$, $C_i\in {\mathbb{H}}^{m\times k}$ for all $i=3,\dots ,8$, and $C_i=-C_i^{*}\in {\mathbb{H}}^{m\times m}$ for $i=1,2$. Assign the following

$$\begin{array}{c}{A}_6=\left[\begin{array}{c}{A}_3\\ -B_3^{*}\end{array}\right],C_9=\left[\begin{array}{c}{C}_3\\ C_4^{*}\end{array}\right],A_7=\left[\begin{array}{c}{A}_4\\ -B_4^{*}\end{array}\right],\hfill \\ C_{10}=\left[\begin{array}{c}{C}_5\\ C_6^{*}\end{array}\right],A_8=\left[\begin{array}{c}{A}_5\\ -B_5^{*}\end{array}\right],C_{11}=\left[\begin{array}{c}{C}_7\end{array}{C}_8^{*}\right],\hfill \\ A_9=A_1{L}_{A_6},B_9=B_1{L}_{A_7},C_{12}=C_1-A_1{X}_{01}{A}_1^{*}-B_1{Y}_{01}{B}_1^{*},\hfill \\ X_{01}=A_6^{†}{C}_9-{\left(A_6^{†}{C}_9\right)}^{*}+A_6^{†}{A}_6{\left(A_6^{†}{C}_9\right)}^{*},\hfill \\ Y_{01}=A_7^{†}{C}_{10}-{\left(A_7^{†}{C}_{10}\right)}^{*}+A_7^{†}{A}_7{\left(A_7^{†}{C}_{10}\right)}^{*},\hfill \\ M_1=R_{A_9}{B}_9,S_1=B_9{L}_{M_1},A_{10}=A_2{L}_{A_8},B_{10}=B_2{L}_{A_7},\hfill \\ C_{13}=C_2-A_2{Z}_{01}{A}_2^{*}-B_2{Y}_{01}{B}_2^{*},\hfill \\ Z_{01}=A_8^{†}{C}_{11}-{\left(A_8{C}_{11}\right)}^{*}+A_8^{†}{A}_8{\left(A_8^{†}{C}_{11}\right)}^{*},\hfill \\ M_2=R_{A_{10}}{B}_{10},S_2=B_{10}{L}_{M_2},\hfill \\ D_1=\left[\begin{array}{cccc}{L}_{M_1}{L}_{S_1}& -L_{B_9}& -L_{M_2}{L}_{S_2}& L_{B_{10}}\end{array}\right],E_1=L_{M_1},F_1=L_{M_2},\hfill \\ G_1=V_{02}-V_{01},V_{02}={\displaystyle \frac{1}{2}}{M}_2^{†}{C}_{13}{\left(B_{10}^{†}\right)}^{*}(I+S_2^{†}{S}_2)-{\displaystyle \frac{1}{2}}(I+S_2^{†}{S}_2)B_{10}^{†}{C}_{13}{\left(M_2^{†}\right)}^{*},\hfill \\ V_{01}={\displaystyle \frac{1}{2}}{M}_1^{†}{C}_{12}{\left(B_9^{†}\right)}^{*}(I+S_1^{†}{S}_1)-{\displaystyle \frac{1}{2}}(I+S_1^{†}{S}_1)B_9^{†}{C}_{12}{\left(M_1^{†}\right)}^{*},\hfill \\ D_2=R_{D_1}{E}_1,E_2=R_{D_1}{F}_1,G_2=R_{D_1}{G}_1{R}_{D_1},M_3=R_{D_2}{E}_2,S_3=E_2{L}_{M_3}.\hfill \end{array}$$

(8)

Then the following conditions are equivalent:

(1)

System (5) is consistent.

(2)

The equalities given below are satisfied:

$$\begin{array}{cc}& R_{A_6}{C}_9=0,A_6{C}_9^{*}=-C_9{A}_6^{*},\hfill \\ \hfill & R_{A_7}{C}_{10}=0,A_7{C}_{10}^{*}=-C_{10}{A}_7^{*},\hfill \\ \hfill & R_{A_8}{C}_{11}=0,A_8{C}_{11}^{*}=-C_{11}{A}_8^{*},\hfill \\ \hfill & R_{A_9}{C}_{12}{R}_{B_9}=0,R_{M_1}{R}_{A_9}{C}_{12}=0,\hfill \\ \hfill & R_{A_{10}}{C}_{13}{R}_{B_{10}}=0,R_{M_2}{R}_{A_{10}}{C}_{13}=0,\hfill \\ \hfill & R_{D_2}{G}_2{R}_{D_2}=0,R_{M_3}{R}_{D_2}{G}_2=0.\hfill \end{array}$$

(9)

(3)

The following rank equalities hold:

$$\begin{array}{cc}\hfill & r\left[\begin{array}{c}{C}_9\\ A_6\end{array}\right]=r\left(A_6\right),A_6{C}_9^{*}=-C_9{A}_6^{*},\hfill \\ \hfill & r\left[\begin{array}{c}{C}_{10}\\ A_7\end{array}\right]=r\left(A_7\right),A_7{C}_{10}^{*}=-C_{10}{A}_7^{*},\hfill \\ \hfill & r\left[\begin{array}{c}{C}_{11}\\ A_8\end{array}\right]=r\left(A_8\right),A_8{C}_{11}^{*}=-C_{11}{A}_8^{*},\hfill \end{array}$$

(10)

$$\begin{array}{cc}\hfill & r\left[\begin{array}{ccc}{C}_1& A_1& B_1{C}_{10}^{*}\\ B_1^{*}& 0& A_7^{*}\\ C_9{A}_1^{*}& A_6& 0\end{array}\right]=r\left[\begin{array}{c}{A}_1\\ A_6\end{array}\right]+r\left[\begin{array}{c}{B}_1\\ A_7\end{array}\right],\hfill \\ \hfill & r\left[\begin{array}{ccc}{C}_1& B_1& A_1\\ C_{10}{B}_1^{*}& A_7& 0\\ C_9{A}_1^{*}& 0& A_6\end{array}\right]=r\left[\begin{array}{cc}{A}_1& B_1\\ 0& A_7\\ A_6& 0\end{array}\right],\hfill \\ \hfill & r\left[\begin{array}{ccc}{C}_2& A_2& B_2{C}_{10}^{*}\\ B_2^{*}& 0& A_7^{*}\\ C_{11}{A}_2^{*}& A_8& 0\end{array}\right]=r\left[\begin{array}{c}{A}_2\\ A_8\end{array}\right]+r\left[\begin{array}{c}{B}_2\\ A_7\end{array}\right],\hfill \\ \hfill & r\left[\begin{array}{ccc}{C}_2& B_2& A_2\\ C_{10}{B}_2^{*}& A_7& 0\\ C_{11}{A}_2^{*}& 0& A_8\end{array}\right]=r\left[\begin{array}{cc}{A}_2& B_2\\ A_8& 0\\ 0& A_7\end{array}\right],\hfill \end{array}$$

(11)

$$\begin{array}{cc}\hfill & r\left[\begin{array}{cccccccccccc}0& 0& B_2^{*}& 0& B_1^{*}& 0& 0& 0& A_7^{*}& 0& 0& 0\\ 0& 0& 0& B_1^{*}& B_1^{*}& 0& 0& 0& 0& A_7^{*}& 0& 0\\ 0& 0& 0& 0& B_1^{*}& B_2^{*}& 0& 0& 0& 0& A_7^{*}& 0\\ 0& 0& B_2^{*}& 0& 0& 0& 0& 0& 0& 0& 0& A_7^{*}\\ B_2& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& -B_1& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ A_7& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& A_7& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\end{array}\right]\hfill \\ \hfill =& r\left[\begin{array}{ccccc}{B}_1& -B_1& -B_1& 0& 0\\ B_2& 0& 0& 0& 0\\ 0& B_2& 0& 0& 0\\ 0& 0& B_1& A_1& 0\\ 0& 0& 0& 0& A_2\\ A_7& 0& 0& 0& 0\\ 0& A_7& 0& 0& 0\\ 0& 0& A_7& 0& 0\\ 0& 0& 0& A_6& 0\\ 0& 0& 0& 0& A_8\end{array}\right]+r\left[\begin{array}{cccc}{B}_1& B_1& 0& 0\\ B_2& 0& 0& 0\\ 0& B_2& A_2& 0\\ 0& 0& 0& A_1\\ A_7& 0& 0& 0\\ 0& A_7& 0& 0\\ 0& 0& A_8& 0\\ 0& 0& 0& A_6\end{array}\right],\hfill \end{array}$$

(12)

$$\begin{array}{cc}\hfill \iff & r\left[\begin{array}{ccccccccc}0& 0& B_1^{*}& B_2^{*}& 0& 0& 0& A_7^{*}& 0\\ 0& 0& B_1^{*}& 0& B_2^{*}& 0& 0& 0& A_7^{*}\\ -B_2& B_2& 0& 0& 0& A_2& 0& 0& 0\\ B_1& 0& 0& 0& 0& 0& A_1& 0& 0\\ 0& B_1& 0& 0& 0& 0& 0& 0& 0\\ A_7& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& A_7& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& A_8& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& A_6& 0& 0\end{array}\right]\hfill \\ \hfill =& r\left[\begin{array}{ccccc}{B}_2& 0& 0& A_2& 0\\ 0& B_1& 0& 0& A_1\\ B_1& B_1& B_1& 0& 0\\ 0& 0& B_2& 0& 0\\ A_7& 0& 0& 0& 0\\ 0& A_7& 0& 0& 0\\ 0& 0& A_7& 0& 0\\ 0& 0& 0& A_8& 0\\ 0& 0& 0& 0& A_6\end{array}\right]+r\left[\begin{array}{cccc}0& B_1& 0& 0\\ B_1& 0& 0& 0\\ 0& B_2& 0& 0\\ 0& 0& A_1& 0\\ B_2& 0& 0& A_2\\ A_7& 0& 0& 0\\ 0& A_7& 0& 0\\ 0& 0& A_6& 0\\ 0& 0& 0& A_8\end{array}\right].\hfill \end{array}$$

(13)

Under these conditions, the general solution to (5) is

$$\begin{array}{cc}\hfill X=& A_6^{†}{C}_9-{\left(A_6^{†}{C}_9\right)}^{*}+A_6^{†}{A}_6{\left(A_6^{†}{C}_9\right)}^{*}+L_{A_6}U{L}_{A_6}^{*},\hfill \\ \hfill Y=& A_7^{†}{C}_{10}-{\left(A_7^{†}{C}_{10}\right)}^{*}+A_7^{†}{A}_7{\left(A_7^{†}{C}_{10}\right)}^{*}+L_{A_7}V{L}_{A_7}^{*},\hfill \\ \hfill Z=& A_8^{†}{C}_{11}-{\left(A_8^{†}{C}_{11}\right)}^{*}+A_8^{†}{A}_8{\left(A_8^{†}{C}_{11}\right)}^{*}+L_{A_8}W{L}_{A_8}^{*},\hfill \end{array}$$

(14)

where

$$\begin{array}{cc}\hfill U=& A_9^{†}{C}_{12}{\left(A_9^{†}\right)}^{*}-{\displaystyle \frac{1}{2}}{A}_9^{†}{B}_9{M}_1^{†}{C}_{12}[I+{\left(B_9^{†}\right)}^{*}{S}_1^{*}]{\left(A_9^{†}\right)}^{*}\hfill \\ \hfill +& {\displaystyle \frac{1}{2}}{A}_1^{†}[I+S_1{B}_9^{†}]C_{12}{\left(M_1^{†}\right)}^{*}{B}_9^{*}{\left(A_9^{†}\right)}^{*}-A_9^{†}{S}_1{U}_1{S}_1^{*}{\left(A_9^{†}\right)}^{*}+L_{A_9}{U}_2-U_2^{*}{L}_{A_9},\hfill \\ \hfill W=& A_{10}^{†}{C}_{13}{\left(A_{10}^{†}\right)}^{*}-{\displaystyle \frac{1}{2}}{A}_{10}^{†}{B}_{10}{M}_2^{†}{C}_{13}[I+{\left(B_{10}^{†}\right)}^{*}{S}_2^{*}]{\left(A_{10}^{†}\right)}^{*}\hfill \\ \hfill +& {\displaystyle \frac{1}{2}}{A}_{10}^{†}[I+S_2{B}_{10}^{†}]C_{13}{M}_2^{*}{B}_{10}^{*}{\left(A_{10}^{†}\right)}^{*}-A_{10}^{†}{S}_2{W}_1{S}_2^{*}{\left(A_{10}^{†}\right)}^{*}+L_{A_{10}}{W}_2-W_2^{*}{L}_{A_{10}},\hfill \end{array}$$

(15)

and the matrix V can be found in two different ways

$$\begin{array}{cc}\hfill V=& {\displaystyle \frac{1}{2}}{M}_1^{†}{C}_{12}{\left(B_9^{†}\right)}^{*}(I+S_1^{†}{S}_1)-{\displaystyle \frac{1}{2}}(I+S_1^{†}{S}_1)B_9^{†}{C}_{12}{\left(M_1^{†}\right)}^{*}+L_{M_1}{U}_1{L}_{M_1}\hfill \\ \hfill +& L_{M_1}{L}_{S_1}{V}_1-V_1^{*}{L}_{S_1}{L}_{M_1}+V_2{L}_{B_9}-L_{B_9}{V}_2^{*},\hfill \\ \hfill V=& {\displaystyle \frac{1}{2}}{M}_2^{†}{C}_{13}{\left(B_{10}^{†}\right)}^{*}(I+S_2^{†}{S}_2)-{\displaystyle \frac{1}{2}}(I+S_2^{†}{S}_2)B_{10}^{†}{C}_{13}{\left(M_2^{†}\right)}^{*}+L_{M_2}{W}_1{L}_{M_2}\hfill \\ \hfill +& L_{M_2}{L}_{S_2}{V}_3-V_3^{*}{L}_{S_2}{L}_{M_2}+V_4{L}_{B_{10}}-L_{B_{10}}{V}_4^{*}.\hfill \end{array}$$

(16)

Here the matrix $V_1$, $V_2$, $V_3$, $V_4$, $U_1$, and $W_1$ are determined by the following. Let

$$T_1^{*}=\left[\begin{array}{cccc}{V}_1^{*}& V_2& V_3^{*}& V_4\end{array}\right].$$

Then,

$$\begin{array}{cc}\hfill T_1=& D_1^{†}(G_1-E_1{U}_1{E}_1^{*}-F_1{T}_2{F}_1^{*})-{\displaystyle \frac{1}{2}}{D}_1^{†}(G_1-E_1{U}_1{E}_1^{*}-F_1{T}_2{F}_1^{*}){\left(D_1^{†}\right)}^{*}{D}_1^{*}\hfill \\ \hfill -& D_1^{†}{T}_4{D}_1^{*}-T_4^{*}{\left(D_1^{†}\right)}^{*}{D}_1^{*}+L_{D_1}{T}_6\hfill \\ \hfill U_1=& D_2^{†}{G}_2{\left(D_2^{†}\right)}^{*}-{\displaystyle \frac{1}{2}}{D}_2^{†}{E}_2{M}_3^{†}{G}_2(I+{\left(E_2^{†}\right)}^{*}{S}_3^{*}){\left(D_2^{†}\right)}^{*}\hfill \\ \hfill +& {\displaystyle \frac{1}{2}}{D}_2^{†}(I+S_3{E}_2^{†})G_2{\left(M_3^{†}\right)}^{*}{E}_2^{*}{\left(D_2^{†}\right)}^{*}-D_2^{†}{S}_3{U}_3{\left(D_2^{†}{S}_3\right)}^{*}+L_{D_2}{U}_4-U_4^{*}{L}_{D_2},\hfill \\ \hfill T_2=-W_1=& {\displaystyle \frac{1}{2}}{M}_3^{†}{G}_2{\left(E_2^{†}\right)}^{*}(I+S_3^{†}{S}_3)H-{\displaystyle \frac{1}{2}}(I+S_3^{†}{S}_3)E_2^{†}{G}_2^{*}{\left(M_3^{†}\right)}^{*}+L_{M_3}{U}_3{L}_{M_3}\hfill \\ \hfill +& L_{M_3}{L}_{S_3}{T}_3-T_3^{*}{L}_{S_3}{L}_{M_3}+T_4{L}_{E_2}-L_{E_2}{T}_4^{*},\hfill \end{array}$$

(17)

where $U_2,W_2,T_3,T_4$, and $T_6={-T}_6^{*}$ are any matrices of acceptable shapes over $\mathbb{H}$.

Proof. 

We write the equations in (6) as follows:

$$\begin{array}{cc}\hfill & A_{3}X=C_3,X{B}_3=C_4,X^{*}=-X,\hfill \\ \hfill & A_{4}Y=C_5,Y{B}_4=C_6,Y^{*}=-Y,\hfill \\ \hfill & A_{1}X{A}_1^{*}+B_{1}Y{B}_1^{*}=C_1,C_1=-C_1^{*}\hfill \end{array}$$

(18)

and

$$\begin{array}{cc}\hfill & A_{4}Y=C_5,Y{B}_4=C_6,Y^{*}=-Y,\hfill \\ \hfill & A_{5}Z=C_7,Z{B}_5=C_8,Z^{*}=-Z,\hfill \\ \hfill & A_{2}Z{A}_2^{*}+B_{2}Y{B}_2^{*}=C_2,C_2=-C_2^{*}.\hfill \end{array}$$

(19)

According to Lemma 2, the general solution to $A_{3}X=C_3,X{B}_3=C_4,X^{*}=-X,$ and $A_{4}Y=C_5,Y{B}_4=C_6,Y^{*}=-Y$ is given by

$$\begin{array}{cc}\hfill X=& A_6^{†}{C}_9-{\left(A_6^{†}{C}_9\right)}^{*}+A_6^{†}{A}_6{\left(A_6^{†}{C}_9\right)}^{*}+L_{A_6}U{L}_{A_6}^{*},\hfill \end{array}$$

(20)

$$\begin{array}{cc}\hfill Y=& A_7^{†}{C}_{10}-{\left(A_7^{†}{C}_{10}\right)}^{*}+A_7^{†}{A}_7{\left(A_7^{†}{C}_{10}\right)}^{*}+L_{A_7}V{L}_{A_7}^{*}\hfill \end{array}$$

(21)

respectively. Using (20) and (21) in the last equation of (18) and making some calculation, we obtain

$$\begin{array}{c}\hfill A_{9}U{A}_9^{*}+B9V{B}_9^{*}=C_{12}.\end{array}$$

(22)

Equation (22) is consistent if and only if

$$\begin{array}{cc}\hfill & R_{A_9}{C}_{12}{R}_{B_9}=0,R_{M_1}{R}_{A_9}{C}_{12}=0.\hfill \end{array}$$

In this case, the general solution to (22) can be written as

$$\begin{array}{cc}\hfill U& =A_9^{†}{C}_{12}{\left(A_9^{†}\right)}^{*}-{\displaystyle \frac{1}{2}}{A}_9^{†}{B}_9{M}_1^{†}{C}_{12}[I+{\left(B_9^{†}\right)}^{*}{S}_1^{*}]{\left(A_9^{†}\right)}^{*}\hfill \\ \hfill & -{\displaystyle \frac{1}{2}}{A}_9^{†}[I+S_1{B}_9^{†}]C_{12}{\left(M_1^{†}\right)}^{*}{B}_9^{*}{\left(A_9^{†}\right)}^{*}-A_9^{†}{S}_1{U}_1{S}_1^{*}{\left(A_9^{†}\right)}^{*}+L_{A_9}{U}_2-U_2^{*}{L}_{A_9},\hfill \\ \hfill V& ={\displaystyle \frac{1}{2}}{M}_1^{†}{C}_{12}{\left(B_9^{†}\right)}^{*}(I+S_1^{†}{S}_1)+{\displaystyle \frac{1}{2}}(I+S_1^{†}{S}_1)B_9^{†}{C}_{12}{\left(M_1^{†}\right)}^{*}\hfill \\ \hfill & +L_{M_1}{U}_1{L}_{M_1}+L_{M_1}{L}_{S_1}{V}_1-V_1^{*}{L}_{S_1}{L}_{M_1}+V_2{L}_{B_9}-L_{B_9}{V}_2^{*}.\hfill \end{array}$$

(23)

Applying Lemma 2 to the system $A_{5}Z=C_7,Z{B}_5=C_8,Y^{*}=-Y$ gives

$$\begin{array}{cc}\hfill Z=& A_8^{†}{C}_{11}-{\left(A_8^{†}{C}_{11}\right)}^{*}+A_8^{†}{A}_8{\left(A_8^{†}{C}_{11}\right)}^{*}+L_{A_8}W{L}_{A_8}^{*}.\hfill \end{array}$$

(24)

Using (21) and (24) in the last equation of (19) and making some calculation, we gain

$$\begin{array}{c}\hfill A_{10}W{A}_{10}^{*}+B_{10}V{B}_{10}^{*}=C_{13}.\end{array}$$

(25)

Equation (25) is solvable if and only if

$$\begin{array}{cc}\hfill & R_{A_{10}}{C}_{13}{R}_{B_{10}}=0,R_{M_2}{R}_{A_{10}}{C}_{13}=0.\hfill \end{array}$$

By Lemma 3 and taking into account equation (1), from this condition it follows that the general solution to (25) can be expressed as

$$\begin{array}{cc}\hfill W& =A_{10}^{†}{C}_{13}{\left(A_{10}^{†}\right)}^{*}-{\displaystyle \frac{1}{2}}{A}_{10}^{†}{B}_{10}{M}_2^{†}{C}_{13}[I+{\left(B_{10}^{†}\right)}^{*}{S}_2^{*}]{\left(A_{10}^{†}\right)}^{*}\hfill \\ \hfill & +{\displaystyle \frac{1}{2}}{A}_{10}^{†}[I+S_2{B}_{10}^{†}]C_{13}(M_2^{*}{B}_2^{*}{\left(A_{10}^{†}\right)}^{*}-A_{10}^{†}{S}_2{W}_1{S}_2^{*}{\left(A_{10}^{†}\right)}^{*}+L_{A_{10}}{W}_2-W_2^{*}{L}_{A_{10}},\hfill \\ \hfill V& ={\displaystyle \frac{1}{2}}{M}_2^{†}{C}_{13}{\left(B_{10}^{†}\right)}^{*}(I+S_2^{†}{S}_2)-{\displaystyle \frac{1}{2}}(I+S_2^{†}{S}_2)B_{10}^{†}{C}_{13}{\left(M_2^{†}\right)}^{*}\hfill \\ \hfill & +L_{M_2}{W}_1{L}_{M_2}+L_{M_2}{L}_{S_2}{V}_3-V_3^{*}{L}_{S_2}{L}_{M_2}+V_4{L}_{B_{10}}-L_{B_{10}}{V}_4^{*},\hfill \end{array}$$

(26)

where $W_2$ a free matrix of adequate shapes over $\mathbb{H}$. The matrices $U_1,W_1,V_1,V_2,V_3,V_4$ are determined as follows.

Denote $T_1^{*}=\left[\begin{array}{cccc}{V}_1^{*}& V_2& V_3^{*}& V_4\end{array}\right],E_1=L_{M_1},F_1=L_{M_2},T_2=-W_1,G_1=V_{02}-V_{01}.$ Equating (23) and (26), we obtain

$$\begin{array}{c}\hfill D_1{T}_1-{\left(D_1{T}_1\right)}^{*}+E_1{U}_1{E_1}^{*}+F_1{T}_2{F}_1^{*}=G_1.\end{array}$$

(27)

By Lemma 3, equation (27) has a solution if and only if the equalities in (9) are satisfied and in this case its general solution can be expressed by (14)-(17).

$\left(2\right)\iff \left(3\right):$ From Lemma 3, we have

$$\begin{array}{cc}\hfill & r\left(R_{A_6}{C}_9\right)=0\iff r\left[\begin{array}{cc}{C}_9& A_6\end{array}\right]=r\left(A_6\right),A_6{C}_9^{*}=-C_9{A}_6^{*},\hfill \\ \hfill & r\left(R_{A_7}{C}_{10}\right)=0\iff r\left[\begin{array}{cc}{C}_{10}& A_7\end{array}\right]=r\left(A_7\right),A_7{C}_{10}^{*}=-C_{10}{A}_7^{*},\hfill \\ \hfill & r\left(R_{A_8}{C}_{11}\right)=0\iff r\left[\begin{array}{cc}{C}_{11}& A_8\end{array}\right]=r\left(A_8\right),A_8{C}_{11}^{*}=-C_{11}{A}_8^{*},\hfill \\ \hfill & r\left(R_{A_9}{C}_{12}{B}_9\right)=0\iff r\left[\begin{array}{cc}{C}_{12}& A_9\\ B_9^{*}& 0\end{array}\right]=r\left(A_9\right)+r\left(B_9\right)\hfill \\ \hfill \iff & r\left[\begin{array}{cc}{C}_1-A_1{X}_{01}{A}_1^{*}-B_1{Y}_{01}{B}_1*& A_1{L}_{A_6}\\ L_{A_7}{B}_1^{*}& 0\end{array}\right]=r\left(A_1{L}_{A_6}\right)+r\left(L_{A_7}{B}_1^{*}\right),\hfill \\ \hfill & r\left[\begin{array}{ccc}{C}_1-A_1{X}_{01}{A}_1^{*}-B_1{Y}_{01}{B}_1*& A_1& 0\\ B_1^{*}& 0& A_7^{*}\\ 0& A_6& 0\end{array}\right]=r\left[\begin{array}{c}{A}_1\\ A_6\end{array}\right]+r\left[\begin{array}{c}{B}_1\\ A_7\end{array}\right],\hfill \\ \hfill & r\left[\begin{array}{ccc}{C}_1& A_1& B_1{C}_{10}^{*}\\ B_1^{*}& 0& A_7^{*}\\ C_9{A}_1^{*}& A_6& 0\end{array}\right]=r\left[\begin{array}{c}{A}_1\\ A_6\end{array}\right]+r\left[\begin{array}{c}{B}_1\\ A_7\end{array}\right],\hfill \\ \hfill & r\left(R_{M_1}{R}_{A_9}{C}_{12}\right)=0\iff r\left[\begin{array}{cc}{R}_{A_9}{C}_{12}& M_1\end{array}\right]=r\left(M_1\right)\iff r\left[\begin{array}{cc}{R}_{A_9}{C}_{12}& R_{A_9}{B}_9\end{array}\right]=r\left(R_{A_9}{B}_9\right),\hfill \\ \hfill & \iff r\left[\begin{array}{ccc}{C}_{12}& B_9& A_9\end{array}\right]=r\left[\begin{array}{cc}{A}_9& B_9\end{array}\right]\hfill \\ \hfill \iff & r\left[\begin{array}{ccc}{C}_1-A_1{X}_{01}{A}_1^{*}-B_1{Y}_{01}{B}_1*& B_1{L}_{A_7}& A_1{L}_{A_6}\end{array}\right]=r\left[\begin{array}{cc}{B}_1{L}_{A_7}& A_1{L}_{A_6}\end{array}\right]\hfill \\ \hfill \iff & r\left[\begin{array}{ccc}{C}_1-A_1{X}_{01}{A}_1^{*}-B_1{Y}_{01}{B}_1*& B_1& A_1\\ 0& A_7& 0\\ 0& 0& A_6\end{array}\right]=r\left[\begin{array}{cc}{B}_1& A_1\\ A_7& 0\\ 0& A_6\end{array}\right]\hfill \\ \hfill \iff & r\left[\begin{array}{ccc}{C}_1& B_1& A_1\\ C_{10}& A_7& 0\\ C_9{A}_1^{*}& 0& A_6\end{array}\right]=r\left[\begin{array}{cc}{B}_1& A_1\\ A_7& 0\\ 0& A_6\end{array}\right],\hfill \\ \hfill & r\left(R_{A_{10}}{C}_{13}{B}_{10}\right)=0\iff r\left[\begin{array}{cc}{C}_{13}& A_{10}\\ B_{10}^{*}& 0\end{array}\right]=r\left(A_{10}\right)+r\left(B_{10}\right)\hfill \\ \hfill \iff & r\left[\begin{array}{cc}{C}_2-A_2{Z}_{01}{A}_2^{*}-B_2{Y}_{01}{B}_2*& A_2{L}_{A_8}\\ L_{A_7}{B}_2^{*}& 0\end{array}\right]=r\left(A_2{L}_{A_8}\right)+r\left(L_{A_7}{B}_2^{*}\right)\hfill \\ \hfill \iff & r\left[\begin{array}{ccc}{C}_2-A_2{Z}_{01}{A}_2^{*}-B_2{Y}_{01}{B}_2*& A_2& 0\\ B_2^{*}& 0& A_7^{*}\\ 0& A_8& 0\end{array}\right]=r\left[\begin{array}{c}{A}_2\\ A_8\end{array}\right]+r\left[\begin{array}{c}{B}_2\\ A_7\end{array}\right]\hfill \\ \hfill \iff & r\left[\begin{array}{ccc}{C}_2& A_2& B_2{C}_{10}^{*}\\ B_2^{*}& 0& A_7^{*}\\ C_{11}{A}_1^{*}& A_8& 0\end{array}\right]=r\left[\begin{array}{c}{A}_2\\ A_8\end{array}\right]+r\left[\begin{array}{c}{B}_2\\ A_7\end{array}\right],\hfill \\ \hfill & r\left(R_{M_2}{R}_{A10}{C}_{13}\right)=0\iff r\left[\begin{array}{cc}{R}_{A_{10}}{C}_{13}& M_2\end{array}\right]=r\left(M_1\right)\hfill \\ \hfill \iff & r\left[\begin{array}{cc}{R}_{A_{10}}{C}_{13}& R_{A_{10}}{B}_{10}\end{array}\right]=r\left(R_{A_{10}}{B}_{10}\right)\hfill \\ \hfill & \iff r\left[\begin{array}{ccc}{C}_{13}& B_{10}& A_{10}\end{array}\right]=r\left[\begin{array}{cc}{A}_{10}& B_{10}\end{array}\right]\hfill \\ \hfill \iff & r\begin{array}{ccc}{C}_2-A_2{Z}_{01}{A}_2^{*}-B_2{Y}_{01}{B}_2^{*}& B_2{L}_{A_7}& A_2{L}_{A_8}\end{array}]=r\left[\begin{array}{cc}{B}_2{L}_{A_7}& A_2{L}_{A_8}\end{array}\right]\hfill \\ \hfill \iff & r\left[\begin{array}{ccc}{C}_2-A_2{Z}_{01}{A}_2^{*}-B_2{Y}_{01}{B}_2^{*}& B_2& A_2\\ 0& A_7& 0\\ 0& 0& A_8\end{array}\right]=r\left[\begin{array}{cc}{A}_2& B_2\\ A_8& 0\\ 0& A_7\end{array}\right]\hfill \\ \hfill \iff & r\left[\begin{array}{ccc}{C}_2& B_2& A_2\\ C_{10}{B}_2^{*}& A_7& 0\\ C_{11}{A}_2^{*}& 0& A_8\end{array}\right]=r\left[\begin{array}{cc}{A}_2& B_2\\ A_8& 0\\ 0& A_7\end{array}\right],\hfill \\ \hfill & R_{D_2}{G}_2{R}_{D_2}^{*}=0\iff r\left[\begin{array}{cc}{G}_2& D_2\\ D_2^{*}& 0\end{array}\right]=r\left(D_2\right)+r\left(E_2\right),\hfill \\ \hfill \iff & r\left[\begin{array}{cc}{R}_{D_1}{G}_1{R}_{D_1}^{*}& R_{D_1}{E}_1\\ F_1^{*}{R}_{D_1}& 0\end{array}\right]=r\left(R_{D_1}{E}_1\right)+r\left(R_{D_1}{F}_1\right)\hfill \\ \hfill \iff & r\left[\begin{array}{ccc}{G}_1& E_1& D_1\\ F_1^{*}& 0& 0\\ D_1^{*}& 0& 0\end{array}\right]=r\left([\begin{array}{cc}{D}_1& E_1\end{array}\right]+r\left[\begin{array}{cc}{D}_1& F_1\end{array}\right]\hfill \\ \hfill \iff & r\left[\begin{array}{cccccc}{v}_{02}-v_{01}& L_{M_1}& L_{M_1}{L}_{S_1}& -L_{B_9}& -L_{M_2}{L}_{S_2}& L_{B_{10}}\\ L_{M_2}& 0& 0& 0& 0& 0\\ L_{S_1}{L}_{M_1}& 0& 0& 0& 0& 0\\ -L_{B_9}& 0& 0& 0& 0& 0\\ L_{S_2}{L}_{M_2}& 0& 0& 0& 0& 0\\ L_{B_{10}}& 0& 0& 0& 0& 0\end{array}\right]\hfill \\ & =r\left[\begin{array}{c}{L}_{M_1}{L}_{M_1}{L}_{S_1}-L_{B_9}-L_{M_2}{L}_{S_2}{L}_{B_{10}}\end{array}\right]+r\left[\begin{array}{c}{L}_{M_2}{L}_{S_1}{L}_{M_1}-L_{B_9}{L}_{S_2}{L}_{M_2}{L}_{B_{10}}\end{array}\right]\hfill \\ \hfill \iff & r\left[\begin{array}{ccccc}{v}_{02}-v_{01}& L_{M_1}& -L_{B_9}& -L_{M_2}{L}_{S_2}& L_{B_{10}}\\ L_{M_2}& 0& 0& 0& 0\\ L_{S_1}{L}_{M_1}& 0& 0& 0& 0\\ -L_{B_9}& 0& 0& 0& 0\\ L_{B_{10}}& 0& 0& 0& 0\end{array}\right]\hfill \\ \hfill =& r\left[\begin{array}{cccc}-I& -L_{M_2}& I& I\\ B_9& 0& 0& 0\\ 0& S_2& 0& 0\\ 0& 0& B_{10}& 0\\ 0& 0& 0& M_1\end{array}\right]+r\left[\begin{array}{cccc}{L}_{M_1}& -I& I& I\\ S_1& 0& 0& 0\\ 0& B_9& 0& 0\\ 0& 0& B_{10}& 0\\ 0& 0& 0& M_2\end{array}\right]\hfill \\ \hfill \iff & r\left[\begin{array}{ccccccccc}{v}_{01}-v_{02}& I& -I& -L_{M_2}& I& 0& 0& 0& 0\\ I& 0& 0& 0& 0& M_2^{*}& 0& 0& 0\\ L_{M_1}& 0& 0& 0& 0& 0& S_1^{*}& 0& 0\\ -I& 0& 0& 0& 0& 0& 0& B_2^{*}& 0\\ I& 0& 0& 0& 0& 0& 0& 0& B_{10}\end{array}\right]\hfill \\ \hfill =& r\left[\begin{array}{cccc}-I& -L_{M_2}& I& I\\ B_9& 0& 0& 0\\ 0& B_1{L}_{M_{10}}& 0& 0\\ 0& 0& B_{10}& 0\\ 0& 0& 0& R_{A_9}{B}_9\end{array}\right]+r\left[\begin{array}{cccc}{L}_{M_1}& -I& I& I\\ B_9{L}_{M_1}& 0& 0& 0\\ 0& B_9& 0& 0\\ 0& 0& B_{10}& 0\\ 0& 0& 0& R_{A_{10}}{B}_{10}\end{array}\right]\hfill \\ \hfill \iff & r\left[\begin{array}{cccccccc}0& 0& B_{10}^{*}& 0& B_9& 0& 0& 0\\ 0& 0& 0& B_9^{*}& B_9& 0& 0& 0\\ 0& 0& 0& 0& B_9& B_{10}^{*}& 0& 0\\ 0& 0& B_{10}^{*}& 0& 0& 0& 0& 0\\ 0& 0& B_{10}^{*}& 0& 0& 0& 0& 0\\ B_{10}& 0& -C_{13}& 0& 0& 0& A_{10}& 0\\ 0& -B_9& 0& -C_{12}^{*}& 0& 0& 0& A_9\end{array}\right]\hfill \\ \hfill =& r\left[\begin{array}{ccccc}{B}_9& -B_9& B_9& 0& 0\\ B_{10}& 0& 0& 0& 0\\ 0& B_{10}& 0& 0& 0\\ 0& 0& B_9& A_9& 0\\ B_{10}& 0& 0& 0& A_{10}\end{array}\right]+r\left[\begin{array}{cccc}{B}_9& B_9& 0& 0\\ B_{10}& 0& 0& 0\\ 0& B_{10}& A_{10}& 0\\ 0& 0& 0& A_9\end{array}\right]\hfill \end{array}$$

$$\begin{array}{cc}\hfill \iff & r\left[\begin{array}{cccccccccccc}0& 0& B_2^{*}& 0& B_1^{*}& 0& 0& 0& A_7^{*}& 0& 0& 0\\ 0& 0& 0& B_1^{*}& B_1^{*}& 0& 0& 0& 0& A_7^{*}& 0& 0\\ 0& 0& 0& 0& B_1^{*}& B_2^{*}& 0& 0& 0& 0& A_7^{*}& 0\\ 0& 0& B_2^{*}& 0& 0& 0& 0& 0& 0& 0& 0& A_7^{*}\\ B_2& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& -B_1& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ A_7& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& A_7& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\end{array}\right]\hfill \\ \hfill =& r\left[\begin{array}{ccccc}{B}_1& -B_1& -B_1& 0& 0\\ B_2& 0& 0& 0& 0\\ 0& B_2& 0& 0& 0\\ 0& 0& B_1& A_1& 0\\ 0& 0& 0& 0& A_2\\ A_7& 0& 0& 0& 0\\ 0& A_7& 0& 0& 0\\ 0& 0& A_7& 0& 0\\ 0& 0& 0& A_6& 0\\ 0& 0& 0& 0& A_8\end{array}\right]+r\left[\begin{array}{cccc}{B}_1& B_1& 0& 0\\ B_2& 0& 0& 0\\ 0& B_2& A_2& 0\\ 0& 0& 0& A_1\\ A_7& 0& 0& 0\\ 0& A_7& 0& 0\\ 0& 0& A_8& 0\\ 0& 0& 0& A_6\end{array}\right],\hfill \\ \hfill & r\left(R_{M_3}{R}_{D_2}{G}_2\right]=0\hfill \\ \hfill \iff & r\left[\begin{array}{cc}{R}_{D_2}{G}_2& M_3\end{array}\right]=r\left(M_3\right)\hfill \\ \hfill \iff & r\left[\begin{array}{cc}{R}_{D_2}{G}_2& R_{D_2}{E}_2\end{array}\right]=r\left(M_3\right)\hfill \\ \hfill \iff & r\left[\begin{array}{ccc}{G}_2& E_2& D_2\end{array}\right]=r\left[\begin{array}{cc}{D}_2& E_2\end{array}\right]\hfill \\ \hfill \iff & r\left[\begin{array}{ccc}{R}_{D_1}{G}_1{R}_{D_1}^{*}& R_{D_1}{F}_1& R_{D_1}{E}_1\end{array}\right]=r\left[\begin{array}{ccc}{F}_1& D_1& E_1\end{array}\right]+r\left(D_1\right)\hfill \\ \hfill \iff & r\left[\begin{array}{ccccccc}{v}_{02}-v_{01}& L_{M_2}& L_{M_1}& L_{M_1}{L}_{S_1}& -L_{B_4}& -L_{M_2}{L}_{S_2}& L_{B_{10}}\\ L_{S_1}{L}_{M_1}& 0& 0& 0& 0& 0& 0\\ -L_{B_9}& 0& 0& 0& 0& 0& 0\\ -L_{S_2}{L}_{M_2}& 0& 0& 0& 0& 0& 0\\ L_{B_{10}}& 0& & 0& 0& 0& 0\end{array}\right]\hfill \\ \hfill =& r\left[\begin{array}{c}{L}_{M_2}{L}_{M_1}{L}_{M_1}{L}_{S_1}-L_{B_9}-L_{M_2}{L}_{S_2}{L}_{B_{10}}\end{array}\right]+r\left[\begin{array}{c}{L}_{M_1}{L}_{S_1}-L_{B_9}-L_{M_2}{L}_{S_2}{L}_{B_{10}}\end{array}\right]\hfill \\ \hfill \iff & r\left[\begin{array}{cccccccccc}{v}_{02}-v_{01}& I& I& -I& -L_{M_2}& I& 0& 0& 0& 0\\ L_{M_1}& 0& 0& 0& 0& 0& S_1^{*}& 0& 0& 0\\ -I& 0& 0& 0& 0& 0& 0& B_9^{*}& 0& 0\\ L_{M_2}& 0& 0& 0& 0& 0& 0& 0& S_2^{*}& 0\\ I& 0& 0& 0& 0& 0& 0& 0& 0& B_{10}^{*}\\ 0& M_2& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& M_1& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& B_9& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& S_2& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& B_{10}& 0& 0& 0& 0\end{array}\right]\hfill \\ \hfill =& r\left[\begin{array}{cccc}{L}_{M_2}& L_{M_1}& -L_{B_9}& L_{B_{10}}\end{array}\right]+r\left[\begin{array}{cccc}{L}_{M_1}{L}_{S_1}& -L_{B_9}& -L_{M_2}{L}_{S_2}& L_{B_{10}}\end{array}\right]\hfill \\ \hfill \iff & r\left[\begin{array}{cccccccccc}{v}_{02}-v_{01}& I& I& -I& -L_{M_2}& I& 0& 0& 0& 0\\ L_{M_1}& 0& 0& 0& 0& 0& L_{M_1}{B}_9^{*}& 0& 0& 0\\ -I& 0& 0& 0& 0& 0& 0& B_9^{*}& 0& 0\\ L_{M_2}& 0& 0& 0& 0& 0& 0& 0& L_{M_2}{B}_{10}^{*}& 0\\ I& 0& 0& 0& 0& 0& 0& 0& 0& B_{10}^{*}\\ 0& R_{A_{10}}{B}_{10}& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& R_{A_9}{B}_9& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& B_9& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& B_{10}{L}_{M_2}& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& B_{10}& 0& 0& 0& 0\end{array}\right]\hfill \\ \hfill =& r\left[\begin{array}{cccc}I& I& -I& I\\ M_2& 0& 0& 0\\ 0& M_1& 0& 0\\ 0& 0& B_9& 0\\ 0& 0& 0& B_{10}\end{array}\right]+r\left[\begin{array}{cccc}{L}_{M_1}& -I& -L_{M_2}& I\\ S_1& 0& 0& 0\\ 0& B_9& 0& 0\\ 0& 0& S_2& 0\\ 0& 0& 0& B_{10}\end{array}\right]\hfill \end{array}$$

Expanding the above equations and using

$$\begin{array}{c}\hfill A_9{U}_{01}{A}_9^{*}+B_9{V}_{01}{B}_9^{*}=C_{12}\\ \hfill A_{10}{W}_{01}{A}_{10}^{*}+B_{10}{V}_{02}{B}_{10}^{*}=C_{13}\end{array}$$

and doing some simplifications in it, we have

$$\begin{array}{cc}\hfill & r\left[\begin{array}{ccccccc}0& 0& -B_9^{*}& B_{10}^{*}& 0& 0& 0\\ 0& 0& -B_9^{*}& 0& B_{10}^{*}& 0& 0\\ -B_{10}& B_{10}& 0& 0& 0& A_{10}& 0\\ B_9& 0& 0& 0& 0& 0& A_9\\ 0& B_9& 0& 0& 0& 0& 0\end{array}\right]\hfill \\ \hfill =& r\left[\begin{array}{ccccc}{B}_{10}& 0& 0& A_{10}& 0\\ 0& B_9& 0& 0& A_9\\ B_9& B_9& B_9& 0& 0\\ 0& 0& B_{10}& 0& 0\end{array}\right]+r\left[\begin{array}{cccc}0& B_9& 0& 0\\ B_9& 0& 0& 0\\ 0& B_{10}& 0& 0\\ 0& 0& A_9& 0\\ B_{10}& 0& 0& A_{10}\end{array}\right]\hfill \\ \hfill \iff & r\left[\begin{array}{ccccccc}0& 0& -L_{A_7}{B}_1^{*}& L_{A_7}{B}_2^{*}& 0& 0& 0\\ 0& 0& -L_{A_7}{B}_1^{*}& 0& L_{A_7}{B}_2^{*}& 0& 0\\ -B_{10}& B_{10}& 0& 0& 0& A_2{L}_{A_8}& 0\\ L_{A_7}{B}_1^{*}& 0& 0& 0& 0& 0& A_1{L}_{A_6}\\ 0& L_{A_7}{B}_1^{*}& 0& 0& 0& 0& 0\end{array}\right]\hfill \\ \hfill =& r\left[\begin{array}{ccccc}{B}_2{L}_{A_7}& 0& 0& A_2{L}_{A_8}& 0\\ 0& B_1{L}_{A_7}& 0& 0& A_9\\ B_1{L}_{A_7}& B_1{L}_{A_7}& B_1{L}_{A_7}& 0& 0\\ 0& 0& B_2{L}_{A_7}& 0& 0\end{array}\right]+r\left[\begin{array}{cccc}0& B_1{L}_{A_7}& 0& 0\\ B_1{L}_{A_7}& 0& 0& 0\\ 0& B_2{L}_{A_7}& 0& 0\\ 0& 0& A_{1}L{A}_6& 0\\ B_2{L}_{A_7}& 0& 0& A_2{L}_{A_8}\end{array}\right]\hfill \\ \hfill \iff & r\left[\begin{array}{ccccccccc}0& 0& B_1^{*}& B_2^{*}& 0& 0& 0& A_7^{*}& 0\\ 0& 0& B_1^{*}& 0& B_2^{*}& 0& 0& 0& A_7^{*}\\ -B_2& B_2& 0& 0& 0& A_2& 0& 0& 0\\ B_1& 0& 0& 0& 0& 0& A_1& 0& 0\\ 0& B_1& 0& 0& 0& 0& 0& 0& 0\\ A_7& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& A_7& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& A_8& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& A_6& 0& 0\end{array}\right]\hfill \\ \hfill =& r\left[\begin{array}{ccccc}{B}_2& 0& 0& A_2& 0\\ 0& B_1& 0& 0& A_1\\ B_1& B_1& B_1& 0& 0\\ 0& 0& B_2& 0& 0\\ A_7& 0& 0& 0& 0\\ 0& A_7& 0& 0& 0\\ 0& 0& A_7& 0& 0\\ 0& 0& 0& A_8& 0\\ 0& 0& 0& 0& A_6\end{array}\right]+r\left[\begin{array}{cccc}0& B_1& 0& 0\\ B_1& 0& 0& 0\\ 0& B_2& 0& 0\\ 0& 0& A_1& 0\\ B_2& 0& 0& A_2\\ A_7& 0& 0& 0\\ 0& A_7& 0& 0\\ 0& 0& A_6& 0\\ 0& 0& 0& A_8\end{array}\right].\hfill \end{array}$$

can be proven similar to the above.    □

Now we discuss a particular case of our system.

If $A_3,A_4,A_5,B_3,B_4,B_5,C_3,C_4,C_5,C_6,C_7$ and $C_8$ are all equal to zero in Theorem 1, then we get the following consequence.

Corollary 1.

Let $A_1\in {\mathbb{C}}^{m\times n}$, $A_2\in {\mathbb{C}}^{m\times q}$, $B_i\in {\mathbb{C}}^{m\times k}$, and $C_i=-C_i^{*}\in {\mathbb{C}}^{m\times m}$ for $i=1,2$. Assign

$$\begin{array}{cc}& M_1=R_{A_1}{B}_1,S_1=B_1{L}_{M_1},M_2=R_{A_2}{B}_2,S_2=B_2{L}_{M_2},A_4=R_{A_3}{L}_{M_1},B_4=R_{A_3}{L}_{M_2},\hfill \\ \hfill & A_3=\left[\begin{array}{cccc}{L}_{B_2}^{*}& -L_{B_1}& L_{M_1}{L}_{S_1}& -L_{M_2}{L}_{S_2}\end{array}\right],M_3=R_{A_4}{B}_4,S_3=B_4{L}_{M_3}\hfill \\ \hfill & C_3=V_{02}-V_{01},V_{02}={\displaystyle \frac{1}{2}}{M}_2^{†}{C}_2{\left(B_2^{†}\right)}^{*}(I+S_2^{†}{S}_2)-{\displaystyle \frac{1}{2}}(I+S_2^{†}{S}_2)B_2^{†}{C}_2{\left(M_2^{†}\right)}^{*},\hfill \\ \hfill & V_{01}={\displaystyle \frac{1}{2}}{M}_1^{†}{C}_1{\left(B_1^{†}\right)}^{*}(I+S_1^{†}{S}_1)-{\displaystyle \frac{1}{2}}(I+S_1^{†}{S}_1)B_1^{†}{C}_1{\left(M_1^{†}\right)}^{*},C_4=R_{A_3}{C}_3{R}_{A_3}.\hfill \end{array}$$

Then the following conditions are equivalent:

(1)

System (5) is consistent.

(2)

The following equalities hold:

$$\begin{array}{cc}& R_{A_1}{C}_1{R}_{B_1}=0,R_{M_1}{R}_{A_1}{C}_1=0,\hfill \\ \hfill & R_{A_2}{C}_2{R}_{B_2}=0,R_{M_2}{R}_{A_2}{C}_2=0,\hfill \\ \hfill & R_{A_4}{C}_4{R}_{B_4}=0,R_{M_3}{R}_{A_4}{C}_4=0.\hfill \end{array}$$

(3)

The following rank equalities hold:

$$\begin{array}{cc}\hfill & r\left[\begin{array}{cc}{C}_1& A_1\\ B_1^{*}& 0\end{array}\right]=r\left(A_1\right)+r\left(B_1\right),r\left[\begin{array}{ccc}{C}_1& B_1& A_1\end{array}\right]=r\left[\begin{array}{cc}{A}_1& B_1\end{array}\right],\hfill \\ \hfill & r\left[\begin{array}{cc}{C}_2& A_2\\ B_2^{*}& 0\end{array}\right]=r\left(A_2\right)+r\left(B_2\right),r\left[\begin{array}{ccc}{C}_2& B_2& A_2\end{array}\right]=r\left[\begin{array}{cc}{A}_2& B_2\end{array}\right],\hfill \\ \hfill \iff & r\left[\begin{array}{ccccccc}0& 0& 0& B_2^{*}& B_1& 0& 0\\ 0& 0& 0& -B_2^{*}& 0& B_1^{*}& 0\\ B_1& 0& 0& 0& 0& C_1& A_1\\ 0& B_2& 0& -C_2& 0& 0& 0\\ -B_2& -B_2& B_2& 0& 0& 0& 0\\ 0& 0& 0& A_2^{*}& 0& 0& 0\end{array}\right]\hfill \\ \hfill =& r\left[\begin{array}{cccc}-B_1& 0& -B_1& A_1\\ B_2& B_2& 0& 0\\ 0& B_1& 0& 0\\ 0& 0& B_2& 0\end{array}\right]+r\left[\begin{array}{cccc}{B}_2& 0& 0& A_2\\ -B_2& B_2& -B_2& 0\\ 0& B_1& 0& 0\\ 0& 0& B_1& 0\end{array}\right],\hfill \\ \hfill & r\left[\begin{array}{cccccccc}0& 0& -B_1^{*}& B_2^{*}& 0& 0& 0& 0\\ 0& 0& -B_1^{*}& 0& B_1^{*}& 0& 0& 0\\ 0& 0& -B_1^{*}& 0& 0& B_2^{*}& 0& 0\\ -B_1^{*}& -B_1^{*}& 0& 0& -C_1& 0& A_1& 0\\ B_2& 0& 0& 0& 0& C_2& A_2& 0\\ 0& B_2& 0& 0& 0& 0& 0& 0\end{array}\right]\hfill \\ \hfill =& r\left[\begin{array}{cccc}-B_1& 0& -B_1& A_1\\ B_2& B_2& 0& 0\\ 0& B_1& 0& 0\\ 0& 0& B_2& 0\end{array}\right]+\left[\begin{array}{cc}{B}_2& 0\\ B_1& 0\\ 0& B_2\end{array}\right]+r\left(B_1\right).\hfill \end{array}$$

Under these conditions, the general solution to (5) is

$$\begin{array}{cc}\hfill X=& A_1^{†}{C}_1{\left(A_1^{†}\right)}^{*}-{\displaystyle \frac{1}{2}}{A}_1^{†}{B}_1{M}_1^{†}{C}_1[I+{\left(B_1^{†}\right)}^{*}{S}_1^{*}]{\left(A_1^{†}\right)}^{*}\hfill \\ \hfill +& {\displaystyle \frac{1}{2}}{A}_1^{†}[I+S_1{B}_1^{†}]C_1{\left(M_1^{†}\right)}^{*}{B}_1^{*}{\left(A_1^{†}\right)}^{*}-A_1^{†}{S}_1{U}_1{S}_1^{*}{\left(A_1^{†}\right)}^{*}+L_{A_1}{V}_1-V_1^{*}{L}_{A_1},\hfill \\ \hfill Z=& A_2^{†}{C}_2{\left(A_2^{†}\right)}^{*}-{\displaystyle \frac{1}{2}}{A}_2^{†}{B}_2{M}_2^{†}{C}_2[I+{\left(B_2^{†}\right)}^{*}{S}_2^{*}]{\left(A_2^{†}\right)}^{*}\hfill \\ \hfill +& {\displaystyle \frac{1}{2}}{A}_2^{†}[I+S_2{B}_2^{†}]C_2{\left(M_2^{†}\right)}^{*}{B}_2^{*}{\left(A_2^{†}\right)}^{*}-A_2^{†}{S}_2{U}_4{S}_2^{*}{\left(A_2^{†}\right)}^{*}+L_{A_2}{V}_2-V_2^{*}{L}_{A_2},\hfill \\ \hfill Y=& {\displaystyle \frac{1}{2}}{M}_1^{†}{C}_1{\left(B_1^{†}\right)}^{*}(I+S_1^{†}{S}_1)-{\displaystyle \frac{1}{2}}(I+S_1^{†}{S}_1)B_1^{†}{C}_1{\left(M_1^{†}\right)}^{*}\hfill \\ \hfill +& L_{M_1}{U}_1{L}_{M_1}+L_{M_1}{L}_{S_1}{U}_2-U_2^{*}{L}_{S_1}{L}_{M_1}+U_3{L}_{B_1}-L_{B_1}{U}_3^{*},\hfill \end{array}$$

or

$$\begin{array}{cc}\hfill Y=& {\displaystyle \frac{1}{2}}{M}_2^{†}{C}_2{\left(B_2^{†}\right)}^{*}(I+S_2^{†}{S}_2)-{\displaystyle \frac{1}{2}}(I+S_2^{†}{S}_2)B_2^{†}{C}_2{\left(M_2^{†}\right)}^{*}\hfill \\ \hfill +& L_{M_2}{U}_4{L}_{M_2}+L_{M_2}{L}_{S_2}{U}_5-U_5^{*}{L}_{S_2}{L}_{M_2}+U_6{L}_{B_2}-L_{B_2}{U}_6^{*},\hfill \end{array}$$

with

$$\begin{array}{cc}\hfill & U_6^{*}=\left[\begin{array}{cccc}{I}_k& 0& 0& 0\end{array}\right]Z,\hfill \\ \hfill & U_3^{*}=\left[\begin{array}{cccc}0& I_k& 0& 0\end{array}\right]Z,\hfill \\ \hfill & U_2=\left[\begin{array}{cccc}0& 0& I_k& 0\end{array}\right]Z,\hfill \\ \hfill & U_5=\left[\begin{array}{cccc}0& 0& 0& I_k\end{array}\right]Z,\hfill \end{array}$$

where

$$\begin{array}{cc}\hfill Z=& A_3^{†}(C_3-L_{M_1}{U}_1{L}_{M_1}-L_{M_2}{U}_4{L}_{M_2})-{\displaystyle \frac{1}{2}}{A}_3^{†}(C_3-L_{M_1}{U}_1{L}_{M_1}-L_{M_2}{U}_4{L}_{M_2})A_3{A}_3^{†}\hfill \\ \hfill -& A_3^{†}{U}_7{A}_3^{*}-U_7^{*}{A}_3{A}_3^{†}+L_{A_3}{U}_8\hfill \\ \hfill U_1=& A_4^{†}{C}_4{\left(A_4^{†}\right)}^{*}-{\displaystyle \frac{1}{2}}{A}_4^{†}{B}_4{M}_3^{†}{C}_4(I+{\left(B_4^{†}\right)}^{*}{S}_3^{*}){\left(A_4^{†}\right)}^{*}\hfill \\ \hfill +& {\displaystyle \frac{1}{2}}{A}_4^{†}(I+S_3{B}_4^{†})C_4{\left(M_3^{†}\right)}^{*}{B}_4^{*}{\left(A_4^{†}\right)}^{*}-A_4^{†}{S}_3{U}_9{\left(A_4^{†}{S}_3\right)}^{*}+L_{A_4}{U}_{10}-U_{10}^{*}{L}_{A_4},\hfill \\ \hfill U_4=& {\displaystyle \frac{1}{2}}{M}_3^{†}{C}_4{\left(B_4^{†}\right)}^{*}(I+S_3^{†}{S}_3)-{\displaystyle \frac{1}{2}}(I+S_3^{†}{S}_3)B_4^{†}{C}_4{\left(M_3^{†}\right)}^{*}+L_{M_3}{U}_{11}{L}_{M_3}\hfill \\ \hfill +& L_{M_3}{L}_{S_3}{U}_{12}-U_{12}^{*}{L}_{S_3}{L}_{M_3}+U_{13}{L}_{B_4}-L_{B_4}{U}_{13}^{*},\hfill \end{array}$$

where  $V_1,V_2,U_7,\dots ,U_{13},U_9={-U}_9^{*},U_{11}=-U_{11}^{*}$ are any matrices of acceptable shapes over $\mathbb{H}$.

3. An algorithm and an example

Following the principles outlined in Theorem 1, we have developed an algorithm for explicitly solving th equations stated in (5). This algorithm makes practical use of MPI representations to construct general solutions, effectively connecting theoretical principles with practical application.

To gain a clearer understanding of the computational aspects of our method, we will refer to a lemma from [44] that provides a method for representing the MPI, an essential component in the solution process.

Lemma 4.

[44][Theorem 4.5] Let $A\in {\mathbb{H}}^{m\times n}$ with $\mathrm{rank}\left(A\right)=r$. Then, the MPI $A^{†}=\left(a_{ij}^{†}\right)\in {\mathbb{H}}^{n\times m}$ can be expressed through the representations stated as

$$\begin{array}{cc}\hfill a_{ij}^{†}=& {\displaystyle \frac{\sum _{\beta \in J_{r,n}\left\{i\right\}}{\mathrm{cdet}}_i{\left({\left(A^{*}A\right)}_{.i}\left(a_{.j}^{*}\right)\right)}_{\beta }^{\beta }}{\sum _{\beta \in J_{r,n}}{\left|A^{*}A\right|}_{\beta }^{\beta }}}={\displaystyle \frac{\sum _{\alpha \in I_{r,m}\left\{j\right\}}{\mathrm{rdet}}_j{\left({\left(A{A}^{*}\right)}_{j.}\left(a_{i.}^{*}\right)\right)}_{\alpha }^{\alpha }}{\sum _{\alpha \in I_{r,m}}{\left|A{A}^{*}\right|}_{\alpha }^{\alpha }}},\hfill \end{array}$$

where ${\mathrm{cdet}}_{i}A$ and ${\mathrm{rdet}}_{i}A$ denote the column and row determinants of $A\in {\mathbb{H}}^{m\times m}$, taken along its i-th column and row, respectively. Furthermore, $A_{\alpha }^{\alpha }$ and $A_{\beta }^{\beta }$ represent principal submatrices of A, while ${\left|A\right|}_{\alpha }^{\alpha }$ and ${\left|A\right|}_{\beta }^{\beta }$ signify principal minors in the sense of row-column determinants when A is Hermitian.

The rows and columns of these submatrices and minors are indexed by $\alpha :=\{{\alpha }_1,\dots ,{\alpha }_r\}\subseteq \{1,\dots ,m\}$ and $\beta :=\{{\beta }_1,\dots ,{\beta }_r\}\subseteq \{1,\dots ,n\},$ where $I_{r,m}:=\{\alpha :1\le {\alpha }_1<\cdots <{\alpha }_r\le m\}$ and $J_{r,n}:=\{\beta :1\le {\beta }_1<\cdots <{\beta }_r\le n\}$. Additionally, $a_{.j}^{*}$ and $a_{i.}^{*}$ refer to the j-th column and the i-th row of $A^{*}$, respectively. The matrices $A_{i.}\left(b\right)$ and $A_{.j}\left(c\right)$ are obtained by replacing the i-th row and j-th column of A with the row vector $b\in {\mathbb{H}}^{1\times n}$ and the column vector $c\in {\mathbb{H}}^m$, respectively.

To find the MPI for any complex matrix by determinantal representations similar to those obtained in Lemma 4, it is enough to change row-column quaternion determinants to an ordinary determinant. Hence, it holds the next.

Lemma 5.

[48][Theorem 2.2] Let $A\in {\mathbb{C}}^{m\times n}$ with $\mathrm{rank}\left(A\right)=r$. Then, the MPI $A^{†}=\left(a_{ij}^{†}\right)\in {\mathbb{C}}^{n\times m}$ can be expressed through the representations stated as

$$\begin{array}{cc}\hfill a_{ij}^{†}=& {\displaystyle \frac{{\displaystyle \sum _{\beta \in J_{r,n}\left\{i\right\}}}{\left|{\left(A^{*}A\right)}_{.i}\left(a_{.j}^{*}\right)\right|}_{\beta }^{\beta }}{{\displaystyle \sum _{\beta \in J_{r,n}}}{\left|A^{*}A\right|}_{\beta }^{\beta }}}={\displaystyle \frac{{\displaystyle \sum _{\alpha \in I_{r,m}\left\{j\right\}}}{\left|{\left(A{A}^{*}\right)}_{j.}\left(a_{i.}^{*}\right)\right|}_{\alpha }^{\alpha }}{{\displaystyle \sum _{\alpha \in I_{r,m}}}{\left|A{A}^{*}\right|}_{\alpha }^{\alpha }}}.\hfill \end{array}$$

Now, we proceed to detail our algorithm for solving the quaternion matrix equations.

Algorithm 1:Finding solutions for the given system

1. Input the matrices $A_i,B_i$ for $i=1,\dots ,5$; $C_i$ for $i=1,\dots ,8$. Ensure they have conformable dimensions over $\mathbb{H}$, and some of them are skew-Hermitian. 2. Compute the requisite matrices as prescribed in (8). 3. Evaluate the consistency of the system using either the matrix equations stated in (9) or rank conditions (10-13). If they do not hold, return “inconsistent". 4. If the consistency conditions are satisfied, compute the matrices $T_i$, $(i=1,2)$, $U_1,U,W,V$ by Eqs. (15)-(17), and by using them compute the solution $X,Y,Z$ as indicated in (14).

To demonstrate the practical application and effectiveness of Algorithm 1, we present the following numerical example. The matrices defined below are employed to find a solution to the equations in (5).

$$\begin{array}{cc}\hfill & A_1=\left[\begin{array}{ccc}3+3\mathbf{i}& 3\mathbf{j}+3\mathbf{k}& 3\mathbf{i}-3\\ 3\mathbf{j}-3\mathbf{k}& 3\mathbf{i}-3& -3\mathbf{j}-3\mathbf{k}\\ -3\mathbf{i}+3& 3\mathbf{j}-3\mathbf{k}& 3\mathbf{i}+3\end{array}\right],A_2=\left[\begin{array}{ccc}3\mathbf{i}& 3\mathbf{j}& 3\mathbf{k}\\ -3& 3\mathbf{k}& -3\mathbf{j}\\ -3\mathbf{j}& 3\mathbf{i}& 3\end{array}\right],A_3=\left[\begin{array}{ccc}1+\mathbf{i}& 1-\mathbf{i}& -1+\mathbf{i}\\ -1+\mathbf{i}& 1+\mathbf{i}& -1-\mathbf{i}\\ 2\mathbf{i}& 2& -2\end{array}\right],\hfill \\ & A_4=\left[\begin{array}{cc}1+\mathbf{j}& 1-\mathbf{j}\\ -1+\mathbf{j}& 1+\mathbf{j}\end{array}\right],A_5=\left[\begin{array}{ccc}1-\mathbf{k}& 1+\mathbf{k}& -1-\mathbf{k}\\ 2& 2\mathbf{k}& -2\mathbf{k}\\ 1+\mathbf{k}& -1+\mathbf{k}& 1-\mathbf{k}\end{array}\right],B_1=\left[\begin{array}{cc}1& \mathbf{j}\\ -\mathbf{k}& \mathbf{i}\\ \mathbf{i}& \mathbf{k}\end{array}\right],B_2=\left[\begin{array}{cc}\mathbf{i}& \mathbf{k}\\ -\mathbf{k}& \mathbf{i}\\ \mathbf{j}& -1\end{array}\right],\hfill \\ & B_3=\left[\begin{array}{ccc}-1+\mathbf{i}& -2& -1-\mathbf{i}\\ -1-\mathbf{i}& -2\mathbf{i}& 1-\mathbf{i}\\ 1+\mathbf{i}& 2\mathbf{i}& 1+\mathbf{i}\end{array}\right],B_4=\left[\begin{array}{cc}-1-\mathbf{j}& -1+\mathbf{j}\\ 1-\mathbf{j}& -1-\mathbf{j}\end{array}\right],B_5=\left[\begin{array}{ccc}\mathbf{i}+\mathbf{j}& 2\mathbf{j}& -\mathbf{i}+\mathbf{j}\\ \mathbf{i}-\mathbf{j}& 2\mathbf{i}& \mathbf{i}+\mathbf{j}\\ -\mathbf{i}+\mathbf{j}& -2\mathbf{i}& -\mathbf{i}-\mathbf{j}\end{array}\right],\hfill \\ & C_1=\left[\begin{array}{ccc}\mathbf{i}& -1+\mathbf{i}& 1-\mathbf{j}\\ 1+\mathbf{i}& \mathbf{j}& \mathbf{k}\\ -1-\mathbf{j}& \mathbf{k}& \mathbf{k}\end{array}\right],C_2=\left[\begin{array}{ccc}\mathbf{k}& -1+\mathbf{j}& 1-\mathbf{k}\\ 1+\mathbf{j}& \mathbf{i}& \mathbf{k}\\ -1-\mathbf{k}& \mathbf{k}& \mathbf{j}\end{array}\right],C_3=\left[\begin{array}{ccc}1+\mathbf{i}& 1-\mathbf{i}& -1+\mathbf{i}\\ -1+\mathbf{i}& 1+\mathbf{i}& -1-\mathbf{i}\\ 2\mathbf{i}& 2& -2\end{array}\right],\hfill \\ & C_4=\left[\begin{array}{ccc}-1+\mathbf{i}& -2& -1-\mathbf{i}\\ -1-\mathbf{i}& -2\mathbf{i}& 1-\mathbf{i}\\ 1+\mathbf{i}& 2\mathbf{i}& -1+\mathbf{i}\end{array}\right],C_5=\left[\begin{array}{cc}1+\mathbf{j}& 1-\mathbf{j}\\ -1+\mathbf{j}& 1+\mathbf{j}\end{array}\right],C_6=\left[\begin{array}{cc}-1-\mathbf{j}& -1+\mathbf{j}\\ 1-\mathbf{j}& -1-\mathbf{j}\end{array}\right],\hfill \\ & C_7=\left[\begin{array}{ccc}1-\mathbf{k}& 1+\mathbf{k}& -1-\mathbf{k}\\ 2& 2\mathbf{k}& -2\mathbf{k}\\ 1+\mathbf{k}& -1+\mathbf{k}& 1-\mathbf{k}\end{array}\right],C_8=\left[\begin{array}{ccc}\mathbf{i}+\mathbf{j}& 2\mathbf{j}& \mathbf{i}-\mathbf{j}\\ \mathbf{i}-\mathbf{j}& 2\mathbf{i}& \mathbf{i}+\mathbf{j}\\ -\mathbf{i}+\mathbf{j}& -2\mathbf{i}& -\mathbf{i}-\mathbf{j}\end{array}\right].\hfill \end{array}$$

Utilizing these matrices, we proceed next with the steps outlined in Algorithm 1 to calculate the solution of the system represented by (5):

• [Step 1] Compute the MPI of the given matrices using Lemma 4. For instance, we have

  $$\begin{array}{cc}\hfill & A_6^{†}={\displaystyle \frac{1}{48}}\left[\begin{array}{cccccc}-1-\mathbf{i}& -1+\mathbf{i}& -2& 1+\mathbf{i}& 2\mathbf{i}& -1+\mathbf{i}\\ 1-\mathbf{i}& -1-\mathbf{i}& -2\mathbf{i}& -1+\mathbf{i}& -2& -1-\mathbf{i}\\ -1+\mathbf{i}& 1+\mathbf{i}& 2\mathbf{i}& 1-\mathbf{i}& 2& 1+\mathbf{i}\end{array}\right].\hfill \end{array}$$
  In particular, after computing the MPI and considering the specific structure of our system, we find that the matrices $S_i$, ($i=1,2,3$), $D_2$, $E_2$, $T_1$, $T_2$, and $U_1$ turn out to be zero matrices. We also consider zero matrices as arbitrary matrices $U_2,W_2,T_3,T_4$, and $T_6={-T}_6^{*}$. This simplification significantly reduces the complexity of the system, facilitating the subsequent steps of our algorithm.
• [Step 2] Validate the given matrices by examining their compatibility with the representation in (9) and (10-13), ensuring the system consistency.
• [Step 3] The next step is to calculate the matrices $U,W$, and V. These matrices are used to construct the general solution to our system. Their calculation relies on values computed earlier and is integral to finalizing the solution according to our algorithm.
• [Step 5] With all necessary matrices computed, we are now in a position to present the solution (14) to the system defined in (5).

  $$\begin{array}{cc}\hfill X& ={\displaystyle \frac{1}{3456}}\left[\begin{array}{ccc}-1731\mathbf{i}-430\mathbf{j}+3\mathbf{k}& 1+2\mathbf{j}+860\mathbf{k}& 1731+\mathbf{j}+430\mathbf{k}\\ -1+2\mathbf{j}+860\mathbf{k}& -3468\mathbf{i}+1720\mathbf{j}-3468\mathbf{k}& -6\mathbf{i}+860\mathbf{j}-2\mathbf{k}\\ -1731+\mathbf{j}+430\mathbf{k}& -6\mathbf{i}+830\mathbf{j}-2\mathbf{k}& -1731\mathbf{i}+430\mathbf{j}-\mathbf{k}\end{array}\right],\hfill \\ \hfill Y& ={\displaystyle \frac{1}{96}}\left[\begin{array}{cc}4\mathbf{i}-47\mathbf{j}+9\mathbf{k}& -49-9\mathbf{i}+4\mathbf{k}\\ 49-9\mathbf{i}+4\mathbf{k}& -4\mathbf{i}-47\mathbf{j}-9\mathbf{k}\end{array}\right],\hfill \\ \hfill Z& ={\displaystyle \frac{1}{6912}}\left[\begin{array}{ccc}-45\mathbf{i}-198\mathbf{j}-2543\mathbf{k}& 1994-15\mathbf{i}-36\mathbf{j}-27\mathbf{k}& -2375+183\mathbf{i}+9\mathbf{j}-27\mathbf{k}\\ -1994-15\mathbf{i}-36\mathbf{j}-27\mathbf{k}& 45\mathbf{i}+174\mathbf{j}-2561\mathbf{k}& 27+9\mathbf{i}+189\mathbf{j}+\mathbf{k}\\ 2375+183\mathbf{i}+9\mathbf{j}-27\mathbf{k}& -27+9\mathbf{i}+189\mathbf{j}+\mathbf{k}& 18\mathbf{i}+6\mathbf{j}+3270\mathbf{k}\end{array}\right].\hfill \end{array}$$

Note that we provide exact numerical values, rather than approximations, because we utilize the direct method for solving the quaternion matrix system (5).

4. Conclusions

This paper establishes the necessary and sufficient conditions for the existence of solutions to a system of anti-Hermitian Sylvester matrix equations with constraints, employing two equivalent approaches: relations expressed using the Moore-Penrose inverse (MPI) and induced projectors of coefficient matrices, and rank conditions. Furthermore, we provide an explicit representation of the anti-Hermitian solution to the system, formulated using the MPI and induced projectors. An algorithm for determining the anti-Hermitian solution, based on this representation, is presented and subsequently demonstrated through a numerical example. This example utilizes techniques for computing row-column determinants, implemented in Maple 2024 with the Clifford package.