$\eta$-Hermitian Solution to a System of Quaternion Matrix Equations

Abstract

For $\eta \in \{{\mathbf {i}},{\mathbf {j}},{\mathbf {k}}\}$, a real quaternion matrix A is said to be $\eta$-Hermitian if $A=A^{\eta *},$ where $A^{\eta *}=-\eta A^{*}\eta$, and $A^{*}$ stands for the conjugate transpose of A. In this paper, we present some practical necessary and sufficient conditions for the existence of an $\eta$-Hermitian solution to a system of constrained two-sided coupled real quaternion matrix equations and provide the general $\eta$-Hermitian solution to the system when it is solvable. Moreover, we present an algorithm and a numerical example to illustrate our results.

1 Introduction

The definition of $\eta$-Hermitian quaternion matrix was first proposed by Took et al. [38] (see Definition 2.1). Nowadays $\eta$-Hermitian quaternion matrices play an important role in widely linear model and statistical signal processing [35,36,37,38].

Since Took et al. [38] first studied the $\eta$-Hermitian quaternion matrix in 2011, there have been some papers to discuss the topics related to $\eta$-Hermitian quaternion matrix. These papers mainly considered the $\eta$-Hermitian solutions to quaternion matrix equations (e.g. [3, 31, 43,44,45,46]) and decompositions of quaternion matrices involving $\eta$-Hermicity (e.g. [15, 17]). For instance, He and Wang [13] provided some necessary and sufficient conditions for the existence of a solution (X, Y, Z) to the quaternion matrix equation

$$\begin{aligned} A_{1}X+(A_{1}X)^{\eta *}+B_{1}YB_{1}^{\eta *}+C_{1}ZC_{1}^{\eta *}=D_{1}, \end{aligned}$$

(1.1)

where Y and Z are required to be $\eta$-Hermitian. They also presented the general solution to equation (1.1) when it is solvable. Liu [29] considered the $\eta$-anti-Hermitian solutions to some classic quaternion matrix equations. Zhang and Wang [47] derived the exact solution to a system of quaternion matrix equations involving $\eta$-Hermicity. Many authors investigated the solutions to quaternion matrix equations (e.g. [1, 2, 4,5,6,7,8,9,10,11,12, 16, 18,19,20,21,22,23,24,25,26,27,28, 32,33,34, 39,40,41, 45]).

In 2014, He and Wang [14] gave the solvability conditions and general $\eta$-Hermitian solution to the system of quaternion matrix equations with two unknowns

$$\begin{aligned} \left\{ \begin{array}{l} A_{1}X=C_{1}, ~XB_{1}=D_{1},\\ A_{2}Y=C_{2}, ~YB_{2}=D_{2},\\ A_{3}XA_{3}^{\eta *}+C_{3}YC_{3}^{\eta *}=P. \end{array} \right. \end{aligned}$$

(1.2)

However, to our knowledge, there has been little information on the generalization of system (1.2). Motivated by the work mentioned above and keeping the wide application of $\eta$-Hermitian quaternion matrices, we consider the $\eta$-Hermitian solution to the system of quaternion matrix equations with three unknowns

$$\begin{aligned} \left\{ \begin{array}{l} A_{1}X=C_{1}, ~XB_{1}=D_{1},\\ A_{2}Y=C_{2}, ~YB_{2}=D_{2},\\ A_{3}Z=C_{3}, ~ZB_{3}=D_{3},\\ A_{4}XA_{4}^{\eta *}+C_{4}YC_{4}^{\eta *}=P,\\ B_{4}ZB_{4}^{\eta *}+D_{4}YD_{4}^{\eta *}=Q, \end{array} \right. \end{aligned}$$

(1.3)

where $A_{i},B_{i},C_{i},D_{i},P=P^{\eta *}$ and $Q=Q^{\eta *}$ are given matrices $(i=1,2,3,4)$.

The remainder of the paper is organized as follows. In Sect. 2, we review the definition and properties of $\eta$-Hermitian quaternion matrix. We also discuss some quaternion matrix equations, which will be used in proving our main result. In Sect. 3, we explore the consistency and $\eta$-Hermitian solutions to system (1.3). In Sect. 4, we provide an algorithm and a numerical example to illustrate our results. Finally, a brief conclusion is given in Sect. 5.

Throughout this paper, let ${\mathbb {R}}$ and ${\mathbb {H}}^{m\times n}$ stand for the real number field and the set of all $m\times n$ matrices over the real quaternion algebra

$$\begin{aligned} {\mathbb {H}}=\big \{a_{0}+a_{1}{\mathbf {i}}+a_{2}{\mathbf {j}} +a_{3}{\mathbf {k}}\big |~{\mathbf {i}}^{2}={\mathbf {j}}^{2} ={\mathbf {k}}^{2}=\mathbf {ijk}=-1,a_{0} ,a_{1},a_{2},a_{3}\in {\mathbb {R}}\big \}, \end{aligned}$$

respectively. The symbol $A^{*}$ stands for the conjugate transpose of matrix A. I and 0 denotes the identity matrix and zero matrix with appropriate sizes, respectively. The rank of $A\in {\mathbb {H}}^{m\times n}$ is defined as the (quaternion) dimension of

$$\begin{aligned} \text {Ran}(A):=\{Ax:~x\in {\mathbb {H}}^{m\times 1}\}, \end{aligned}$$

the range of A (Definition 3.2.3 in [32]). If $A\in {\mathbb {H}}^{m\times n},$ then there exist nonsingular matrices P and Q such that

$$\begin{aligned} PAQ=\begin{pmatrix}I_{r(A)}&{}\quad 0\\ 0&{}\quad 0\end{pmatrix}, \end{aligned}$$

where the symbol r(A) is the rank of A. The Moore-Penrose inverse $A^{\dag }$ of a quaternion matrix A, is defined to be the unique matrix $A^{\dag },$ such that

$$\begin{aligned} \text {(i)}~AA^{\dag }A=A,~\text {(ii)}~A^{\dag }AA^{\dag }=A^{\dag },~\text {(iii)} ~(AA^{\dag })^{*}=AA^{\dag },~\text {(iv)}~(A^{\dag }A)^{*}=A^{\dag }A. \end{aligned}$$

Furthermore, $L_{A}$ and $R_{A}$ stand for the projectors $L_{A} =I-A^{\dag }A$ and $R_{A}=I-AA^{\dag }$ induced by A, respectively.

2 Preliminaries

In this section, we first review the definition of $\eta$-Hermitian quaternion matrix.

Definition 2.1

($\eta$-Hermitian quaternion matrix) [38] For $\eta \in \{{\mathbf {i}},{\mathbf {j}},{\mathbf {k}}\}$, a square real quaternion matrix A is said to be $\eta$-Hermitian if $A=A^{\eta *},$ where $A^{\eta *}=-\eta A^{*}\eta$.

Some basic properties of $\eta$-Hermitian matrix are given in the following lemma.

Lemma 2.1

[13] Let $A\in {\mathbb {H}}^{m\times n}$ be given. Then,

• (1) $(A^{\eta *})^{\dag }=(A^{\dag })^{\eta *}$.

• (2) $r(A)=r(A^{\eta *})$.

• (3) $(L_{A})^{\eta *}=R_{A^{\eta *}}$.

• (4) $(R_{A})^{\eta *}=L_{A^{\eta *}}$.

Now, we discuss some quaternion matrix equations, which will be used for proving the main result of the paper.

Lemma 2.2

[14] The system of quaternion matrix equations

$$\begin{aligned} A_{1}X=C_{1},~XB_{1}=D_{1} \end{aligned}$$

(2.1)

has an $\eta$-Hermitian solution if and only if $R_{E_{1}}F_{1}=0$ and $E_{1}F_{1}^{\eta *}=F_{1}E_{1}^{\eta *}$, where

$$\begin{aligned} E_{1}= \begin{bmatrix} A_{1}\\ B_{1}^{\eta *} \end{bmatrix} , ~F_{1}= \begin{bmatrix} C_{1}\\ D_{1}^{\eta *} \end{bmatrix}. \end{aligned}$$

In this case, the general $\eta$-Hermitian solution to (2.1) can be expressed as

$$\begin{aligned} X=E_{1}^{\dag }F_{1}+F_{1}^{\eta *}(E_{1}^{\dag })^{\eta *} -E_{1}^{\dag }E_{1}F_{1}^{\eta *}(E_{1}^{\dag })^{\eta *}+L_{E_{1}}W(L_{E_{1}})^{\eta *}, \end{aligned}$$

where W is an arbitrary $\eta$-Hermitian matrix over ${\mathbb {H}}$ with appropriate size.

Lemma 2.3

[13] Consider the quaternion matrix equation (1.1). Set

$$\begin{aligned} A=R_{A_{1}}B_{1},~B=R_{A_{1}}C_{1},~ C=R_{A_{1}}D_{1}(R_{A_{1}})^{\eta *},~M_{1}=R_{A}B,~S_{1}=BL_{M_{1}}. \end{aligned}$$

Then, the following statements are equivalent:

• (1) Equation (1.1) has a solution (X, Y, Z), where Y and Z are $\eta$-Hermitian.

• (2)
  $$\begin{aligned} R_{M_{1}}R_{A}C=0,\qquad R_{A}CL_{B^{\eta *}}=0. \end{aligned}$$
• (3)
  $$\begin{aligned} r\begin{bmatrix}D_{1} &{} C_{1} &{} B_{1} &{} A_{1}\\ A_{1}^{\eta *} &{} 0 &{} 0 &{} 0\end{bmatrix}=r[C_{1},~B_{1},~A_{1}]+r(A_{1}), \end{aligned}$$
  $$\begin{aligned} r\begin{bmatrix}D_{1} &{} B_{1} &{} A_{1}\\ A_{1}^{\eta *} &{} 0 &{} 0\\ C_{1}^{\eta *} &{} 0 &{} 0\end{bmatrix}=r[B_{1},~A_{1}]+r[A_{1},~C_{1}]. \end{aligned}$$

  If matrix equation (1.1) is consistent, then the general solution to (1.1) is

  $$\begin{aligned} X=&\,A_{1}^{\dag }(D_{1}-B_{1}YB_{1}^{\eta *}-C_{1}ZC_{1}^{\eta *}) -\frac{1}{2}A_{1}^{\dag }(D_{1}-B_{1}YB_{1}^{\eta *} -C_{1}ZC_{1}^{\eta *})(A_{1}^{\dag })^{\eta *}A_{1}^{\eta *}\\&+L_{A_{1}}U_{1}+U_{2}^{\eta *}(A_{1}^{\dag })^{\eta *}A_{1}^{\eta *} -A_{1}^{\dag }U_{2}A_{1}^{\eta *},\\ Y=&\,Y^{\eta *}=A^{\dag }C(A^{\dag })^{\eta *}-\frac{1}{2}A^{\dag }BM_{1}^{\dag }C[I+(B^{\dag })^{\eta *}S_{1}^{\eta *}](A^{\dag })^{\eta *}\\&-\frac{1}{2}A^{\dag }(I+S_{1}B^{\dag })C(M_{1}^{\dag })^{\eta *} B^{\eta *}(A^{\dag })^{\eta *}\\&-A^{\dag }S_{1}VS_{1}^{\eta *}(A^{\dag })^{\eta *}+L_{A}U_{3}+(L_{A}U_{3})^{\eta *},\\ Z=&\,Z^{\eta *}=\frac{1}{2}M_{1}^{\dag }C(B^{\dag })^{\eta *}[I+(S_{1}^{\dag }S_{1})^{\eta *}]+\frac{1}{2}(I+S_{1}^{\dag }S_{1})B^{\dag }C(M_{1}^{\dag })^{\eta *} +L_{M_{1}}V(L_{M_{1}})^{\eta *}\\&+L_{B}U_{4}+(L_{B}U_{4})^{\eta *} +L_{M_{1}}L_{S_{1}}U_{5}+(L_{M_{1}}L_{S_{1}}U_{5})^{\eta *}, \end{aligned}$$

  where $U_{1},U_{2},U_{3},U_{4},U_{5},V=V^{\eta *}$ are arbitrary matrices over ${\mathbb {H}}$ with appropriate sizes.

The following lemma can be easily generalized to ${\mathbb {H}}$.

Lemma 2.4

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

• (1) $r(A)+r(R_{A}B)=r(B)+r(R_{B}A)=r [ A,~ B ]$.

• (2) $r(A)+r(CL_{A})=r(C)+r(AL_{C})=r \begin{bmatrix} A\\ C \end{bmatrix}$.

3 The General $\eta$-Hermitian Solution to System (1.3)

In this section, we derive some practical necessary and sufficient conditions for the existence of an $\eta$-Hermitian solution (X, Y, Z) to the system of quaternion matrix equations

$$\begin{aligned} \left\{ \begin{array}{l} A_{1}X=C_{1}, ~XB_{1}=D_{1},\\ A_{2}Y=C_{2}, ~YB_{2}=D_{2},\\ A_{3}Z=C_{3}, ~ZB_{3}=D_{3},\\ A_{4}XA_{4}^{\eta *}+C_{4}YC_{4}^{\eta *}=P,\\ B_{4}ZB_{4}^{\eta *}+D_{4}YD_{4}^{\eta *}=Q. \end{array} \right. \end{aligned}$$

(3.1)

Moreover, we also give the expression of the general $\eta$-Hermitian solution to system (3.1) when it is solvable. For simplicity, we denote

$$\begin{aligned} E_{j}=&\, \begin{bmatrix} A_{j}\\ B_{j}^{\eta *} \end{bmatrix}, ~F_{j}= \begin{bmatrix} C_{j}\\ D_{j}^{\eta *} \end{bmatrix},~(j=1,2,3), \end{aligned}$$

(3.2)

$$\begin{aligned} A_{5}=&\,A_{4}L_{E_{1}},~B_{5}=B_{4}L_{E_{3}},~C_{5}=C_{4} L_{E_{2}},~D_{5}=D_{4}L_{E_{2}}, \end{aligned}$$

(3.3)

$$\begin{aligned} P_{1}=&\,P-A_{4}[E_{1}^{\dag }F_{1}+F_{1}^{\eta *}(E_{1}^{\dag })^{\eta *} -E_{1}^{\dag }E_{1}F_{1}^{\eta *}(E_{1}^{\dag })^{\eta *}]A_{4}^{\eta *}\nonumber \\&-C_{4}[E_{2}^{\dag }F_{2}+F_{2}^{\eta *} (E_{2}^{\dag })^{\eta *}-E_{2}^{\dag }E_{2}F_{2}^{\eta *}(E_{2}^{\dag })^{\eta *}]C_{4}^{\eta *}, \end{aligned}$$

(3.4)

$$\begin{aligned} Q_{1}=&\,Q-B_{4}[E_{3}^{\dag }F_{3}+F_{3}^{\eta *}(E_{3}^{\dag })^{\eta *} -E_{3}^{\dag }E_{3}F_{3}^{\eta *}(E_{3}^{\dag })^{\eta *}]B_{4}^{\eta *}\nonumber \\&-D_{4}[E_{2}^{\dag }F_{2}+F_{2}^{\eta *} (E_{2}^{\dag })^{\eta *}-E_{2}^{\dag }E_{2}F_{2}^{\eta *} (E_{2}^{\dag })^{\eta *}]D_{4}^{\eta *}, \end{aligned}$$

(3.5)

$$\begin{aligned} M_{1}=&\,R_{A_{5}}C_{5},~S_{1}=C_{5}L_{M_{1}},~M_{2}=R_{B_{5}}D_{5}, ~S_{2}=D_{5}L_{M_{2}}, \end{aligned}$$

(3.6)

$$\begin{aligned} A_{6}=&\,[L_{C_{5}},~L_{M_{1}}L_{S_{1}},~-L_{D_{5}},~-L_{M_{2}}L_{S_{2}}], \end{aligned}$$

(3.7)

$$\begin{aligned} P_{2}=&\,\frac{1}{2}M_{2}^{\dag }Q_{1}(D_{5}^{\dag })^{\eta *} [I+(S_{2}^{\dag }S_{2})^{\eta *}]+\frac{1}{2}(I+S_{2}^{\dag }S_{2})D_{5}^{\dag }Q_{1}(M_{2}^{\dag })^{\eta *}\nonumber \\&-\frac{1}{2}M_{1}^{\dag }P_{1}(C_{5}^{\dag })^{\eta *} [I+(S_{1}^{\dag }S_{1})^{\eta *}]-\frac{1}{2}(I+S_{1}^{\dag }S_{1})C_{5}^{\dag }P_{1}(M_{1}^{\dag })^{\eta *}, \end{aligned}$$

(3.8)

$$\begin{aligned} A=&\,R_{A_{6}}L_{M_{1}},~B=R_{A_{6}}L_{M_{2}},~C=R_{A_{6}}P_{2} (R_{A_{6}})^{\eta *},~M=R_{A}B,~S=BL_{M}. \end{aligned}$$

(3.9)

Next, we establish the main result of the paper.

Theorem 3.1

Let $A_{i},B_{i},C_{i},D_{i},P=P^{\eta *}$ and $Q=Q^{\eta *}$ be given over ${\mathbb {H}}$, $(\eta \in \{{\mathbf {i}},{\mathbf {j}},{\mathbf {k}}\},i=1,\ldots ,4)$. Then, the following statements are equivalent:

• (1) System (3.1) has an $\eta$-Hermitian solution (X, Y, Z).

• (2)
  $$\begin{aligned} R_{E_{j}}F_{j}=&\,0,~E_{j}F_{j}^{\eta *}=F_{j}E_{j}^{\eta *},~(j=1,2,3), \end{aligned}$$

  (3.10)
  $$\begin{aligned} R_{M_{1}}R_{A_{5}}P_{1}=&\,0,~R_{A_{5}}P_{1}(R_{C_{5}})^{\eta *}=0,~ R_{M_{2}}R_{B_{5}}Q_{1}=0,~R_{B_{5}}Q_{1}(R_{D_{5}})^{\eta *}=0, \end{aligned}$$

  (3.11)
  $$\begin{aligned} R_{M}R_{A}C=&\,0,~R_{A}C(R_{B})^{\eta *}=0. \end{aligned}$$

  (3.12)
• (3)
  $$\begin{aligned}&r\begin{bmatrix}A_{j}&{}C_{j}\\ B_{1}^{\eta *}&{}D_{1}^{\eta *}\end{bmatrix} =r\begin{bmatrix}A_{j}\\ B_{1}^{\eta *}\end{bmatrix},~ \begin{bmatrix} A_{j}C_{j}^{\eta *}&{}A_{j}D_{j}\\ B_{j}^{\eta *}C_{j}^{\eta *}&{}B_{j}^{\eta *}D_{j} \end{bmatrix}\nonumber \\&\quad =\begin{bmatrix} C_{j}A_{j}^{\eta *}&{}C_{j}B_{j}\\ D_{j}^{\eta *}A_{j}^{\eta *}&{}D_{j}^{\eta *}B_{j} \end{bmatrix}, \quad (j=1,2,3), \end{aligned}$$

  (3.13)
  $$\begin{aligned}&r\begin{bmatrix} P&{}A_{4}&{}C_{4}\\ C_{1}A_{4}^{\eta *}&{}A_{1}&{}0\\ D_{1}^{\eta *}A_{4}^{\eta *}&{}B_{1}^{\eta *}&{}0\\ C_{2}C_{4}^{\eta *}&{}0&{}A_{2}\\ D_{2}^{\eta *}C_{4}^{\eta *}&{}0&{}B_{2}^{\eta *}\end{bmatrix}= r\begin{bmatrix} A_{4}&{}C_{4}\\ A_{1}&{}0\\ B_{1}^{\eta *}&{}0\\ 0&{}A_{2}\\ 0&{}B_{2}^{\eta *}\end{bmatrix}, \end{aligned}$$

  (3.14)
  $$\begin{aligned}&r\begin{bmatrix} P&{}A_{4}&{}C_{4}C_{2}^{\eta *}&{}C_{4}D_{2}\\ C_{4}^{\eta *}&{}0&{}A_{2}^{\eta *}&{}B_{2}\\ C_{1}A_{4}^{\eta *}&{}A_{1}&{}0&{}0\\ D_{1}^{\eta *}A_{4}^{\eta *}&{}B_{1}^{\eta *}&{}0&{}0\end{bmatrix}= r\begin{bmatrix} A_{1}\\ B_{1}^{\eta *}\\ A_{4}\end{bmatrix}+r\begin{bmatrix} A_{2}\\ B_{2}^{\eta *}\\ C_{4}\end{bmatrix}, \end{aligned}$$

  (3.15)
  $$\begin{aligned}&r\begin{bmatrix} Q&{}B_{4}&{}D_{4}\\ C_{3}B_{4}^{\eta *}&{}A_{3}&{}0\\ D_{3}^{\eta *}B_{4}^{\eta *}&{}B_{3}^{\eta *}&{}0\\ C_{2}D_{4}^{\eta *}&{}0&{}A_{2}\\ D_{2}^{\eta *}D_{4}^{\eta *}&{}0&{}B_{2}^{\eta *}\end{bmatrix}= r\begin{bmatrix} B_{4}&{}D_{4}\\ A_{3}&{}0\\ B_{3}^{\eta *}&{}0\\ 0&{}A_{2}\\ 0&{}B_{2}^{\eta *}\end{bmatrix}, \end{aligned}$$

  (3.16)
  $$\begin{aligned}&r\begin{bmatrix} Q&{}B_{4}&{}D_{4}C_{2}^{\eta *}&{}D_{4}D_{2}\\ D_{4}^{\eta *}&{}0&{}A_{2}^{\eta *}&{}B_{2}\\ C_{3}B_{4}^{\eta *}&{}A_{3}&{}0&{}0\\ D_{3}^{\eta *}B_{4}^{\eta *}&{}B_{3}^{\eta *}&{}0&{}0\end{bmatrix}= r\begin{bmatrix} A_{3}\\ B_{3}^{\eta *}\\ B_{4}\end{bmatrix}+r\begin{bmatrix} A_{2}\\ B_{2}^{\eta *}\\ D_{4}\end{bmatrix}, \end{aligned}$$

  (3.17)
  $$\begin{aligned}&r\begin{bmatrix} 0&{}-C_{4}^{\eta *}&{}D_{4}^{\eta *}&{}0&{}0&{}A_{2}^{\eta *}&{}B_{2}\\ C_{4}&{}P&{}0&{}A_{4}&{}0&{}0&{}0\\ D_{4}&{}0&{}Q&{}0&{}B_{4}&{}D_{4}C_{2}^{\eta *}&{}D_{4}D_{2}\\ A_{2}&{}C_{2}C_{4}^{\eta *}&{}0&{}0&{}0&{}0&{}0\\ B_{2}^{\eta *}&{}D_{2}^{\eta *}C_{4}^{\eta *}&{}0&{}0&{}0&{}0&{}0\\ 0&{}C_{1}A_{4}^{\eta *}&{}0&{}A_{1}&{}0&{}0&{}0\\ 0&{}D_{1}^{\eta *}A_{4}^{\eta *}&{}0&{}B_{1}^{\eta *}&{}0&{}0&{}0\\ 0&{}0&{}C_{3}B_{4}^{\eta *}&{}0&{}A_{3}&{}0&{}0\\ 0&{}0&{}D_{3}^{\eta *}B_{4}^{\eta *}&{}0&{}B_{3}^{\eta *}&{}0&{}0 \end{bmatrix}=r\begin{bmatrix}C_{4}\\ D_{4}\\ A_{2}\\ B_{2}^{\eta *} \end{bmatrix}\nonumber \\&\qquad + r\begin{bmatrix}C_{4}&{}A_{4}&{}0\\ D_{4}&{}0&{}B_{4}\\ A_{2}&{}0&{}0\\ B_{2}^{\eta *}&{}0&{}0\\ 0&{}A_{1}&{}0\\ 0&{}B_{1}^{\eta *}&{}0\\ 0&{}0&{}A_{3}\\ 0&{}0&{}B_{3}^{\eta *}\end{bmatrix}, \end{aligned}$$

  (3.18)
  $$\begin{aligned}&r\begin{bmatrix} 0&{}-C_{4}^{\eta *}&{}D_{4}^{\eta *}&{}0&{}A_{2}^{\eta *}&{}B_{2}&{}0&{}0\\ C_{4}&{}P&{}0&{}A_{4}&{}0&{}0&{}0&{}0\\ D_{4}&{}0&{}Q&{}0&{}D_{4}C_{2}^{\eta *}&{}D_{4}D_{2}&{}B_{4}C_{3}^{\eta *}&{}B_{4}D_{3}\\ A_{2}&{}C_{2}C_{4}^{\eta *}&{}0&{}0&{}0&{}0&{}0&{}0\\ B_{2}^{\eta *}&{}D_{2}^{\eta *}C_{4}^{\eta *}&{}0&{}0&{}0&{}0&{}0&{}0\\ 0&{}C_{1}A_{4}^{\eta *}&{}0&{}A_{1}&{}0&{}0&{}0&{}0\\ 0&{}D_{1}^{\eta *}A_{4}^{\eta *}&{}0&{}B_{1}^{\eta *}&{}0&{}0&{}0&{}0\\ 0&{}0&{}B_{4}^{\eta *}&{}0&{}0&{}0&{}A_{3}^{\eta *}&{}B_{3} \end{bmatrix}= r\begin{bmatrix}C_{4}&{}A_{4}\\ D_{4}&{}0\\ A_{2}&{}0\\ B_{2}^{\eta *}&{}0\\ 0&{}A_{1}\\ 0&{}B_{1}^{\eta *}\end{bmatrix}\nonumber \\&\qquad + r\begin{bmatrix} C_{4}&{}0\\ D_{4}&{}B_{4}\\ A_{2}&{}0\\ B_{2}^{\eta *}&{}0\\ 0&{}A_{3}\\ 0&{}B_{3}^{\eta *}\end{bmatrix}. \end{aligned}$$

  (3.19)

If system (3.1) is consistent, the general $\eta$-Hermitian solution to (3.1) can be expressed as

$$\begin{aligned} X=&\,E_{1}^{\dag }F_{1}+F_{1}^{\eta *}(E_{1}^{\dag })^{\eta *} -E_{1}^{\dag }E_{1}F_{1}^{\eta *}(E_{1}^{\dag })^{\eta *}+L_{E_{1}}X_{1}(L_{E_{1}})^{\eta *}, \end{aligned}$$

(3.20)

$$\begin{aligned} Y=&\,E_{2}^{\dag }F_{2}+F_{2}^{\eta *}(E_{2}^{\dag })^{\eta *} -E_{2}^{\dag }E_{2}F_{2}^{\eta *}(E_{2}^{\dag })^{\eta *}+L_{E_{2}}Y_{1}(L_{E_{2}})^{\eta *}, \end{aligned}$$

(3.21)

$$\begin{aligned} Z=&\,E_{3}^{\dag }F_{3}+F_{3}^{\eta *}(E_{3}^{\dag })^{\eta *} -E_{3}^{\dag }E_{3}F_{3}^{\eta *}(E_{3}^{\dag })^{\eta *}+L_{E_{3}}Z_{1}(L_{E_{3}})^{\eta *}, \end{aligned}$$

(3.22)

where

$$\begin{aligned} X_{1}=&\,A_{5}^{\dag }P_{1}(A_{5}^{\dag })^{\eta *}-\frac{1}{2} A_{5}^{\dag }C_{5}M_{1}^{\dag }P_{1}[I+(C_{5}^{\dag })^{\eta *}S_{1}^{\eta *}](A_{5}^{\dag })^{\eta *}\\&-\frac{1}{2}A_{5}^{\dag }(I+S_{1}C_{5}^{\dag })P_{1}(M_{1}^{\dag })^{\eta *}C_{5}^{\eta *}(A_{5}^{\dag })^{\eta *}\\&-A_{5}^{\dag }S_{1}V_{1}S_{1}^{\eta *}(A_{5}^{\dag })^{\eta *} +L_{A_{5}}U_{1}+(L_{A_{5}}U_{1})^{\eta *},\\ Z_{1}=&\,B_{5}^{\dag }Q_{1}(B_{5}^{\dag })^{\eta *} -\frac{1}{2}B_{5}^{\dag }D_{5}M_{2}^{\dag }Q_{1}[I+(D_{5}^{\dag })^{\eta *}S_{2}^{\eta *}](B_{5}^{\dag })^{\eta *}\\&-\frac{1}{2}B_{5}^{\dag }(I+S_{2}D_{5}^{\dag })Q_{1}(M_{2}^{\dag })^{\eta *}D_{5}^{\eta *}(B_{5}^{\dag })^{\eta *}\\&+B_{5}^{\dag }S_{2}V_{2}S_{2}^{\eta *}(B_{5}^{\dag })^{\eta *} +L_{B_{5}}U_{4}+(L_{B_{5}}U_{4})^{\eta *},\\ Y_{1}=&\,\frac{1}{2}M_{1}^{\dag }P_{1}(C_{5}^{\dag })^{\eta *} [I+(S_{1}^{\dag }S_{1})^{\eta *}]+\frac{1}{2}(I+S_{1}^{\dag }S_{1})C_{5}^{\dag }P_{1}(M_{1}^{\dag })^{\eta *} +L_{M_{1}}V_{1}(L_{M_{1}})^{\eta *}\\&+L_{C_{5}}U_{2}+(L_{C_{5}}U_{2})^{\eta *} +L_{M_{1}}L_{S_{1}}U_{3}+(L_{M_{1}}L_{S_{1}}U_{3})^{\eta *}, \end{aligned}$$

or

$$\begin{aligned} Y_{1}=&\,\frac{1}{2}M_{2}^{\dag }Q_{1}(D_{5}^{\dag })^{\eta *} [I+(S_{2}^{\dag }S_{2})^{\eta *}]+\frac{1}{2}(I+S_{2}^{\dag }S_{2})D_{5}^{\dag }Q_{1}(M_{2}^{\dag })^{\eta *}\\&-L_{M_{2}}V_{2}(L_{M_{2}})^{\eta *}\\&+L_{D_{5}}U_{5}+(L_{D_{5}}U_{5})^{\eta *} +L_{M_{2}}L_{S_{2}}U_{6}+(L_{M_{2}}L_{S_{2}}U_{6})^{\eta *},\\ U_{2}=&\,[I_{m},~0,~0,~0]\Bigg [A_{6}^{\dag }[P_{2} -L_{M_{1}}V_{1}(L_{M_{1}})^{\eta *}-L_{M_{2}}V_{2}(L_{M_{2}})^{\eta *}]\\&+L_{A_{6}}W_{1}+W_{2}^{\eta *}(A_{6}^{\dag })^{\eta *}(A_{6})^{\eta *}\\&-\frac{1}{2}A_{6}^{\dag }[P_{2}-L_{M_{1}}V_{1}(L_{M_{1}})^{\eta *} -L_{M_{2}}V_{2}(L_{M_{2}})^{\eta *}](A_{6}^{\dag })^{\eta *}A_{6}^{\eta *} -A_{6}^{\dag }W_{2}A_{6}^{\eta *}\Bigg ],\\ U_{3}=&\,[0,~I_{m},~0,~0]\Bigg [A_{6}^{\dag }[P_{2}-L_{M_{1}} V_{1}(L_{M_{1}})^{\eta *}-L_{M_{2}}V_{2}(L_{M_{2}})^{\eta *}]\\&+L_{A_{6}}W_{1}+W_{2}^{\eta *}(A_{6}^{\dag })^{\eta *}(A_{6})^{\eta *}\\&-\frac{1}{2}A_{6}^{\dag }[P_{2}-L_{M_{1}}V_{1}(L_{M_{1}})^{\eta *} -L_{M_{2}}V_{2}(L_{M_{2}})^{\eta *}](A_{6}^{\dag })^{\eta *}A_{6}^{\eta *} -A_{6}^{\dag }W_{2}A_{6}^{\eta *}\Bigg ],\\ U_{5}=&\,[0,~0,~I_{m},~0]\Bigg [A_{6}^{\dag }[P_{2} -L_{M_{1}}V_{1}(L_{M_{1}})^{\eta *}-L_{M_{2}}V_{2}(L_{M_{2}})^{\eta *}]\\&+L_{A_{6}}W_{1}+W_{2}^{\eta *}(A_{6}^{\dag })^{\eta *}(A_{6})^{\eta *}\\&-\frac{1}{2}A_{6}^{\dag }[P_{2}-L_{M_{1}}V_{1}(L_{M_{1}})^{\eta *} -L_{M_{2}}V_{2}(L_{M_{2}})^{\eta *}](A_{6}^{\dag })^{\eta *}A_{6}^{\eta *} -A_{6}^{\dag }W_{2}A_{6}^{\eta *}\Bigg ],\\ U_{6}=&\,[0,~0,~0,~I_{m}]\Bigg [A_{6}^{\dag }[P_{2}-L_{M_{1}} V_{1}(L_{M_{1}})^{\eta *}-L_{M_{2}}V_{2}(L_{M_{2}})^{\eta *}]\\&+L_{A_{6}}W_{1}+W_{2}^{\eta *}(A_{6}^{\dag })^{\eta *}(A_{6})^{\eta *}\\&-\frac{1}{2}A_{6}^{\dag }[P_{2}-L_{M_{1}}V_{1}(L_{M_{1}})^{\eta *} -L_{M_{2}}V_{2}(L_{M_{2}})^{\eta *}](A_{6}^{\dag })^{\eta *} A_{6}^{\eta *}-A_{6}^{\dag }W_{2}A_{6}^{\eta *}\Bigg ],\\ V_{1}=&\,V_{1}^{\eta *}=A^{\dag }C(A^{\dag })^{\eta *} -\frac{1}{2}A^{\dag }BM^{\dag }C[I+(B^{\dag })^{\eta *}S^{\eta *}](A^{\dag })^{\eta *}\\&-\frac{1}{2}A^{\dag }(I+SB^{\dag })C(M^{\dag })^{\eta *}B^{\eta *}(A^{\dag })^{\eta *} \\&-A^{\dag }SW_{3}S^{\eta *}(A^{\dag })^{\eta *}+L_{A}W_{4}+(L_{A}W_{4})^{\eta *},\\ V_{2}=&\,V_{2}^{\eta *} = \frac{1}{2}M^{\dag }C(B^{\dag })^{\eta *}[I+(S^{\dag }S)^{\eta *}]+\frac{1}{2}(I+S^{\dag }S)B^{\dag }C(M^{\dag })^{\eta *}+L_{M}W_{3}(L_{M})^{\eta *}\\&+L_{B}W_{5}+(L_{B}W_{5})^{\eta *} +L_{M}L_{S}W_{6}+(L_{M}L_{S}W_{6})^{\eta *} , \end{aligned}$$

and $U_{1},U_{4},W_{1},W_{2},W_{3}=W_{3}^{\eta *},W_{4},W_{5},W_{6}$ are arbitrary matrices over ${\mathbb {H}}$ with appropriate sizes, m is the column number of $A_{2}$.

Proof

$(1)\Longleftrightarrow (2):$ It follows from Lemma 2.2 that the systems

$$\begin{aligned} A_{1}X=&\,C_{1}, ~XB_{1}=D_{1}, \end{aligned}$$

(3.23)

$$\begin{aligned} A_{2}Y=&\,C_{2}, ~YB_{2}=D_{2} \end{aligned}$$

(3.24)

and

$$\begin{aligned} A_{3}Z=C_{3}, ~ZB_{3}=D_{3} \end{aligned}$$

(3.25)

have an $\eta$-Hermitian solution, respectively, if and only if

$$\begin{aligned} R_{E_{1}}F_{1}=&\,0,~E_{1}F_{1}^{\eta *}=F_{1}E_{1}^{\eta *},\\ R_{E_{2}}F_{2}=&\,0,~E_{2}F_{3}^{\eta *}=F_{2}E_{2}^{\eta *},\\ R_{E_{3}}F_{3}=&\,0,~E_{3}F_{3}^{\eta *}=F_{3}E_{3}^{\eta *}, \end{aligned}$$

where $E_{j}$ and $F_{j}$ are defined in (3.2). In this case, the general $\eta$-Hermitian solutions to systems (3.23)–(3.25) can be expressed as

$$\begin{aligned} X=E_{1}^{\dag }F_{1}+F_{1}^{\eta *}(E_{1}^{\dag })^{\eta *} -E_{1}^{\dag }E_{1}F_{1}^{\eta *}(E_{1}^{\dag })^{\eta *}+L_{E_{1}}X_{1}(L_{E_{1}})^{\eta *}, \end{aligned}$$

(3.26)

$$\begin{aligned} Y=E_{2}^{\dag }F_{2}+F_{2}^{\eta *}(E_{2}^{\dag })^{\eta *} -E_{2}^{\dag }E_{2}F_{2}^{\eta *}(E_{2}^{\dag })^{\eta *}+L_{E_{2}}Y_{1}(L_{E_{2}})^{\eta *}, \end{aligned}$$

(3.27)

$$\begin{aligned} Z=E_{3}^{\dag }F_{3}+F_{3}^{\eta *}(E_{3}^{\dag })^{\eta *} -E_{3}^{\dag }E_{3}F_{3}^{\eta *}(E_{3}^{\dag })^{\eta *}+L_{E_{3}}Z_{1}(L_{E_{3}})^{\eta *}, \end{aligned}$$

(3.28)

where $X_{1},Y_{1}$ and $Z_{1}$ are arbitrary $\eta$-Hermitian matrices over ${\mathbb {H}}$ with appropriate sizes. Substituting (3.26)–(3.28) into

$$\begin{aligned} \left\{ \begin{array}{c} A_{4}XA_{4}^{\eta *}+C_{4}YC_{4}^{\eta *}=P,\\ B_{4}ZB_{4}^{\eta *}+D_{4}YD_{4}^{\eta *}=Q, \end{array} \right. \end{aligned}$$

yields

$$\begin{aligned} \left\{ \begin{array}{c} A_{5}X_{1}A_{5}^{\eta *}+C_{5}Y_{1}C_{5}^{\eta *}=P_{1},\\ B_{5}Z_{1}B_{5}^{\eta *}+D_{5}Y_{1}D_{5}^{\eta *}=Q_{1}, \end{array} \right. \end{aligned}$$

(3.29)

where $A_{5},B_{5},C_{5},D_{5},P_{1}$ and $Q_{1}$ are defined in (3.3)–(3.5).

Now, we want to find the solvability conditions and general $\eta$-Hermitian solution to system (3.29). From Lemma 2.3, it follows that the first equation in system (3.29) has an $\eta$-Hermitian solution if and only if

$$\begin{aligned} R_{M_{1}}R_{A_{5}}P_{1}=0,~R_{A_{5}}P_{1}(R_{C_{5}})^{\eta *}=0, \end{aligned}$$

where $M_{1}$ is defined in (3.6). The general $\eta$-Hermitian solution to the first equation in system (3.29) can be expressed as

$$\begin{aligned} X_{1}=&\,A_{5}^{\dag }P_{1}(A_{5}^{\dag })^{\eta *} -\frac{1}{2}A_{5}^{\dag }C_{5}M_{1}^{\dag }P_{1}[I+(C_{5}^{\dag })^{\eta *}S_{1}^{\eta *}](A_{5}^{\dag })^{\eta *}\nonumber \\&-\frac{1}{2}A_{5}^{\dag }(I+S_{1}C_{5}^{\dag })P_{1}(M_{1}^{\dag })^{\eta *}C_{5}^{\eta *}(A_{5}^{\dag })^{\eta *}\nonumber \\&-A_{5}^{\dag }S_{1}V_{1}S_{1}^{\eta *}(A_{5}^{\dag })^{\eta *} +L_{A_{5}}U_{1}+(L_{A_{5}}U_{1})^{\eta *},\nonumber \\ Y_{1}=&\,\frac{1}{2}M_{1}^{\dag }P_{1}(C_{5}^{\dag })^{\eta *} [I+(S_{1}^{\dag }S_{1})^{\eta *}]+\frac{1}{2}(I+S_{1}^{\dag } S_{1})C_{5}^{\dag }P_{1}(M_{1}^{\dag })^{\eta *}+L_{M_{1}}V_{1}(L_{M_{1}})^{\eta *}\nonumber \\&+L_{C_{5}}U_{2}+(L_{C_{5}}U_{2})^{\eta *} +L_{M_{1}}L_{S_{1}}U_{3}+(L_{M_{1}}L_{S_{1}}U_{3})^{\eta *}, \end{aligned}$$

(3.30)

where $S_{1}$ is defined in (3.6), $U_{1},U_{2},U_{3}$ and $V_{1}=V_{1}^{\eta *}$ are arbitrary matrices over ${\mathbb {H}}$ with appropriate size.

Similarly, the second equation in system (3.29) has an $\eta$-Hermitian solution if and only if

$$\begin{aligned} R_{M_{2}}R_{B_{5}}Q_{1}=0,~R_{B_{5}}Q_{1}(R_{D_{5}})^{\eta *}=0, \end{aligned}$$

where $M_{2}$ is defined in (3.6). The general $\eta$-Hermitian solution to the second equation in system (3.29) can be expressed as

$$\begin{aligned} Z_{1}=&\,B_{5}^{\dag }Q_{1}(B_{5}^{\dag })^{\eta *}-\frac{1}{2}B_{5}^{\dag } D_{5}M_{2}^{\dag }Q_{1}[I+(D_{5}^{\dag })^{\eta *}S_{2}^{\eta *}](B_{5}^{\dag })^{\eta *}\nonumber \\&-\frac{1}{2}B_{5}^{\dag }(I+S_{2}D_{5}^{\dag })Q_{1}(M_{2}^{\dag })^{\eta *}D_{5}^{\eta *}(B_{5}^{\dag })^{\eta *}\nonumber \\&+B_{5}^{\dag }S_{2}V_{2}S_{2}^{\eta *}(B_{5}^{\dag })^{\eta *} +L_{B_{5}}U_{4}+(L_{B_{5}}U_{4})^{\eta *},\nonumber \\ Y_{1}=&\,\frac{1}{2}M_{2}^{\dag }Q_{1}(D_{5}^{\dag })^{\eta *} [I+(S_{2}^{\dag }S_{2})^{\eta *}]+\frac{1}{2}(I+S_{2}^{\dag } S_{2})D_{5}^{\dag }Q_{1}(M_{2}^{\dag })^{\eta *}-L_{M_{2}}V_{2}(L_{M_{2}})^{\eta *} \nonumber \\&+L_{D_{5}}U_{5}+(L_{D_{5}}U_{5})^{\eta *} +L_{M_{2}}L_{S_{2}}U_{6}+(L_{M_{2}}L_{S_{2}}U_{6})^{\eta *}, \end{aligned}$$

(3.31)

where $S_{2}$ is defined in (3.6), $U_{4},U_{5},U_{6}$ and $V_{2}=V_{2}^{\eta *}$ are arbitrary matrices over ${\mathbb {H}}$ with appropriate sizes.

Equating (3.30) and (3.31), we have

$$\begin{aligned}&[L_{C_{5}},~L_{M_{1}}L_{S_{1}},~-L_{D_{5}},~ -L_{M_{2}}L_{S_{2}}]\begin{bmatrix}U_{2}\\ U_{3}\\ U_{5}\\ U_{6}\end{bmatrix}\\&\qquad + \left( [L_{C_{5}},~L_{M_{1}}L_{S_{1}},~-L_{D_{5}},~ -L_{M_{2}}L_{S_{2}}]\begin{bmatrix}U_{2}\\ U_{3}\\ U_{5}\\ U_{6}\end{bmatrix}\right) ^{\eta *}\\&\qquad +L_{M_{1}}V_{1}(L_{M_{1}})^{\eta *}+L_{M_{2}}V_{2}(L_{M_{2}})^{\eta *}\\&\quad =\frac{1}{2}M_{2}^{\dag }Q_{1}(D_{5}^{\dag })^{\eta *} [I+(S_{2}^{\dag }S_{2})^{\eta *}]+\frac{1}{2}(I+S_{2}^{\dag }S_{2})D_{5}^{\dag }Q_{1}(M_{2}^{\dag })^{\eta *}\\&\qquad -\frac{1}{2}M_{1}^{\dag }P_{1}(C_{5}^{\dag })^{\eta *} [I+(S_{1}^{\dag }S_{1})^{\eta *}]-\frac{1}{2}(I+S_{1}^{\dag }S_{1})C_{5}^{\dag }P_{1}(M_{1}^{\dag })^{\eta *}. \end{aligned}$$

Then, it can be rewritten as

$$\begin{aligned} A_{6}\begin{bmatrix}U_{2}\\ U_{3}\\ U_{5}\\ U_{6}\end{bmatrix}+\left( A_{6}\begin{bmatrix}U_{2}\\ U_{3}\\ U_{5}\\ U_{6}\end{bmatrix}\right) ^{\eta *} +L_{M_{1}}V_{1}(L_{M_{1}})^{\eta *}+L_{M_{2}}V_{2}(L_{M_{2}})^{\eta *}=P_{2}, \end{aligned}$$

(3.32)

where $V_{1}$ and $V_{2}$ are $\eta$-Hermitian, $A_{6}$ and $P_{2}$ are defined in (3.7) and (3.8), separately. We want to find the general solution to equation (3.32), where $V_{1}$ and $V_{2}$ are $\eta$-Hermitian. By Lemma 2.3, we know that equation (3.32) has a solution if and only if

$$\begin{aligned} R_{M}R_{A}C=0,~R_{A}C(R_{B})^{\eta *}=0, \end{aligned}$$

where A, B, C, M are defined in (3.9). In this case, the general solution to equation (3.32) is given as follows:

$$\begin{aligned} \begin{bmatrix}U_{2}\\ U_{3}\\ U_{5}\\ U_{6}\end{bmatrix}=&\,A_{6}^{\dag }[P_{2}-L_{M_{1}} V_{1}(L_{M_{1}})^{\eta *}-L_{M_{2}}V_{2}(L_{M_{2}})^{\eta *}]\\&-\frac{1}{2}A_{6}^{\dag }[P_{2}-L_{M_{1}}V_{1}(L_{M_{1}})^{\eta *} -L_{M_{2}}V_{2}(L_{M_{2}})^{\eta *}](A_{6}^{\dag })^{\eta *}A_{6}^{\eta *}\\&+L_{A_{6}}W_{1}+W_{2}^{\eta *}(A_{6}^{\dag })^{\eta *}(A_{6})^{\eta *} -A_{6}^{\dag }W_{2}A_{6}^{\eta *},\\ V_{1}=&\,V_{1}^{\eta *}=A^{\dag }C(A^{\dag })^{\eta *}\\&-\frac{1}{2}A^{\dag }BM^{\dag }C[I+(B^{\dag })^{\eta *}S^{\eta *}](A^{\dag })^{\eta *} -\frac{1}{2}A^{\dag }(I+SB^{\dag })C(M^{\dag })^{\eta *}B^{\eta *}(A^{\dag })^{\eta *} \\&-A^{\dag }SW_{3}S^{\eta *}(A^{\dag })^{\eta *}+L_{A}W_{4}+(L_{A}W_{4})^{\eta *},\\ V_{2}=&\,V_{2}^{\eta *} = \frac{1}{2}M^{\dag }C(B^{\dag })^{\eta *}[I+(S^{\dag }S)^{\eta *}]+\frac{1}{2}(I+S^{\dag }S)B^{\dag }C(M^{\dag })^{\eta *}+L_{M}W_{3}(L_{M})^{\eta *}\\&+L_{B}W_{5}+(L_{B}W_{5})^{\eta *} +L_{M}L_{S}W_{6}+(L_{M}L_{S}W_{6})^{\eta *} , \end{aligned}$$

where $W_{1},W_{2},W_{3}=W_{3}^{\eta *},W_{4},W_{5},W_{6}$ are arbitrary matrices over ${\mathbb {H}}$ with appropriate sizes.

$(2)\Longleftrightarrow (3):$ By Lemma 2.4 and block Gaussian elimination, we obtain

$$\begin{aligned} R_{E_{j}}F_{j}=0\Longleftrightarrow r[E_{j},~F_{j}]=r(E_{j}), \end{aligned}$$

i.e. (3.10) $\Longleftrightarrow$ (3.13). It follows from Lemma 2.2 that systems (3.23)–(3.25) have an $\eta$-Hermitian solution, respectively, under the conditions (3.10). Note that

$$\begin{aligned} X_{0}:=&\,E_{1}^{\dag }F_{1}+F_{1}^{\eta *}(E_{1}^{\dag })^{\eta *} -E_{1}^{\dag }E_{1}F_{1}^{\eta *}(E_{1}^{\dag })^{\eta *},\\ Y_{0}:=&\,E_{2}^{\dag }F_{2}+F_{2}^{\eta *}(E_{2}^{\dag })^{\eta *} -E_{2}^{\dag }E_{2}F_{2}^{\eta *}(E_{2}^{\dag })^{\eta *}, \end{aligned}$$

and

$$\begin{aligned} Z_{0}:=E_{3}^{\dag }F_{3}+F_{3}^{\eta *}(E_{3}^{\dag })^{\eta *} -E_{3}^{\dag }E_{3}F_{3}^{\eta *}(E_{3}^{\dag })^{\eta *} \end{aligned}$$

are special $\eta$-Hermitian solutions to systems (3.23)–(3.25). Then, $P_{1}$ and $Q_{1}$ can be rewritten as follows:

$$\begin{aligned} P_{1}=&\,P-A_{4}X_{0}A_{4}^{\eta *}-C_{4}Y_{0}C_{4}^{\eta *}, \end{aligned}$$

(3.33)

$$\begin{aligned} Q_{1}=&\,Q-B_{4}Z_{0}B_{4}^{\eta *}-D_{4}Y_{0}D_{4}^{\eta *}. \end{aligned}$$

(3.34)

Now, we will use Lemma 2.4, (3.33), (3.34) and block Gaussian elimination to prove (3.11) $\Longleftrightarrow$ (3.14)–(3.17). Applying Lemma 2.4 to $R_{M_{1}}R_{A_{5}}P_{1}=0$ yields

$$\begin{aligned}&R_{M_{1}}R_{A_{5}}P_{1}=\,0 \Leftrightarrow r(R_{M_{1}}R_{A_{5}}P_{1})=0 \Leftrightarrow r[P_{1},~A_{5},~C_{5}]=r[A_{5},~C_{5}]\\&\Leftrightarrow r[P_{1},~A_{4}L_{E_{1}},~C_{4}L_{E_{2}}]=\,r[A_{4}L_{E_{1}},~C_{4}L_{E_{2}}] \Leftrightarrow r\begin{bmatrix}P_{1}&{}A_{4}&{}C_{4}\\ 0&{}E_{1}&{}0\\ 0&{}0&{}E_{2}\end{bmatrix} =r\begin{bmatrix}A_{4}&{}C_{4}\\ E_{1}&{}0\\ 0&{}E_{2}\end{bmatrix}\\&\Leftrightarrow r\begin{bmatrix}P-A_{4}X_{0}A_{4}^{\eta *}-C_{4}Y_{0} C_{4}^{\eta *}&{}A_{4}&{}C_{4}\\ 0&{}E_{1}&{}0\\ 0&{}0&{}E_{2}\end{bmatrix} =r\begin{bmatrix}A_{4}&{}C_{4}\\ E_{1}&{}0\\ 0&{}E_{2}\end{bmatrix}\\&\Leftrightarrow r\begin{bmatrix}P&{}A_{4}&{}C_{4}\\ E_{1}X_{0}A_{4}^{\eta *} &{}E_{1}&{}0\\ E_{2}Y_{0}C_{4}^{\eta *}&{}0&{}E_{2}\end{bmatrix}= r\begin{bmatrix}A_{4}&{}C_{4}\\ E_{1}&{}0\\ 0&{}E_{2}\end{bmatrix}\\&\Leftrightarrow r\begin{bmatrix}P&{}A_{4}&{}C_{4}\\ F_{1}A_{4}^{\eta *}&{}E_{1}&{}0\\ F_{2}C_{4}^{\eta *}&{}0&{}E_{2}\end{bmatrix}=r\begin{bmatrix}A_{4} &{}C_{4}\\ E_{1}&{}0\\ 0&{}E_{2}\end{bmatrix} \Leftrightarrow (3.14). \end{aligned}$$

Similarly, it can be shown that

$$\begin{aligned} R_{A_{5}}P_{1}(R_{C_{5}})^{\eta *}=&\,0\Leftrightarrow (3.15),\\ R_{M_{2}}R_{B_{5}}Q_{1}=&\,0\Leftrightarrow (3.16),\\ R_{B_{5}}Q_{1}(R_{D_{5}})^{\eta *}=&\,0\Leftrightarrow (3.17). \end{aligned}$$

Next, we prove that (3.12) $\Longleftrightarrow$ (3.18) and (3.19). Note that

$$\begin{aligned} Y_{01}:=\frac{1}{2}M_{1}^{\dag }P_{1}(C_{5}^{\dag })^{\eta *} [I+(S_{1}^{\dag }S_{1})^{\eta *}]+\frac{1}{2}(I+S_{1}^{\dag }S_{1})C_{5}^{\dag }P_{1}(M_{1}^{\dag })^{\eta *} \end{aligned}$$

and

$$\begin{aligned} Y_{02}:=\frac{1}{2}M_{2}^{\dag }Q_{1}(D_{5}^{\dag })^{\eta *} [I+(S_{2}^{\dag }S_{2})^{\eta *}]+\frac{1}{2}(I+S_{2}^{\dag }S_{2})D_{5}^{\dag }Q_{1}(M_{2}^{\dag })^{\eta *} \end{aligned}$$

are special solutions to the first two equations in system (3.29), separately, under the conditions (3.11). Then, $P_{2}$ can be rewritten as follows

$$\begin{aligned} P_{2}=Y_{02}-Y_{01}. \end{aligned}$$

(3.35)

We next use Lemma 2.4, together with (3.29), (3.33), (3.34), (3.35) and block Gaussian elimination to show $R_{M}R_{A}C=0$ $\Longleftrightarrow$ (3.18). Applying Lemma 2.4 to $R_{M}R_{A}C=0$ gives

$$\begin{aligned}&R_{M}R_{A}C=0 \Leftrightarrow r(R_{M}R_{A}C)=0\\&\qquad \Leftrightarrow r\begin{bmatrix}P_{2}&{}A_{6}&{}L_{M_{1}}&{}L_{M_{2}}\\ A_{6}^{\eta *}&{}0&{}0&{}0\end{bmatrix}= r[A_{6},~L_{M_{1}},~L_{M_{2}}]+r(A_{6}) \\&\qquad \Leftrightarrow r\begin{bmatrix}P_{2}&{}L_{M_{1}}&{}L_{M_{2}}\\ R_{S_{1}^{\eta *}} R_{M_{1}^{\eta *}}&{}0&{}0\\ R_{S_{2}^{\eta *}}R_{M_{2}^{\eta *}}&{}0&{}0\end{bmatrix}= r[L_{M_{1}},~L_{M_{2}}]+r\begin{bmatrix}R_{S_{1}^{\eta *}} R_{M_{1}^{\eta *}}\\ R_{S_{2}^{\eta *}}R_{M_{2}^{\eta *}}\end{bmatrix} \\&\qquad \Leftrightarrow r\begin{bmatrix}P_{2}&{}I&{}I&{}0&{}0&{}0&{}0\\ I&{}0&{}0&{}C_{5}^{\eta *} &{}0&{}0&{}0\\ I&{}0&{}0&{}0&{}D_{5}^{\eta *}&{}0&{}0\\ 0&{}C_{5}&{}0&{}0&{}0&{}A_{5}&{}0\\ 0&{}0&{}D_{5}&{}0&{}0&{}0&{}B_{5}\end{bmatrix}= r\begin{bmatrix}I&{}I&{}0&{}0\\ C_{5}&{}0&{}A_{5}&{}0\\ 0&{}D_{5}&{}0&{}B_{5}\end{bmatrix}+r\begin{bmatrix}I&{}C_{5}^{\eta *}&{}0\\ I&{}0&{}D_{5}^{\eta *}\end{bmatrix}\\&\qquad \Leftrightarrow r\begin{bmatrix}Y_{02}-Y_{01}&{}I&{}I&{}0&{}0&{}0&{}0\\ I&{}0&{}0&{}C_{5}^{\eta *}&{}0&{}0&{}0\\ I&{}0&{}0&{}0&{}D_{5}^{\eta *}&{}0&{}0\\ 0&{}C_{5}&{}0&{}0&{}0&{}A_{5}&{}0\\ 0&{}0&{}D_{5}&{}0&{}0&{}0&{}B_{5}\end{bmatrix}= r\begin{bmatrix}I&{}I&{}0&{}0\\ C_{5}&{}0&{}A_{5}&{}0\\ 0&{}D_{5}&{}0&{}B_{5} \end{bmatrix}+r\begin{bmatrix}I&{}C_{5}^{\eta *}&{}0\\ I&{}0&{}D_{5}^{\eta *}\end{bmatrix}\\&\qquad \Leftrightarrow r\begin{bmatrix}0&{}C_{5}^{\eta *}&{}D_{5}^{\eta *}&{}0&{}0\\ C_{5}&{}-P_{1}&{}0&{}A_{5}&{}0\\ D_{5}&{}0&{}Q_{1}&{}0&{}B_{5}\end{bmatrix}=r\begin{bmatrix} C_{5}&{}A_{5}&{}0\\ D_{5}&{}0&{}B_{5}\end{bmatrix}+r\begin{bmatrix}C_{5}\\ D_{5}\end{bmatrix}\\&\qquad \Leftrightarrow r\begin{bmatrix} 0&{}C_{4}^{\eta *}&{}D_{4}^{\eta *}&{}0&{}0&{}E_{2}^{\eta *}\\ C_{4}&{}-P_{1}&{}0&{}A_{4}&{}0&{}0\\ D_{4}&{}0&{}Q_{1}&{}0&{}B_{4}&{}0\\ E_{2}&{}0&{}0&{}0&{}0&{}0\\ 0&{}0&{}0&{}E_{1}&{}0&{}0\\ 0&{}0&{}0&{}0&{}E_{3}&{}0 \end{bmatrix}=r\begin{bmatrix}C_{4}&{}A_{4}&{}0\\ D_{4}&{}0&{}B_{4}\\ E_{2}&{}0&{}0\\ 0&{}E_{1}&{}0\\ 0&{}0&{}E_{3}\end{bmatrix}+r \begin{bmatrix}C_{4}\\ D_{4}\\ E_{2}\end{bmatrix}\\&\qquad \Leftrightarrow r\begin{bmatrix} 0&{}C_{4}^{\eta *}&{}D_{4}^{\eta *}&{}0&{}0&{}E_{2}^{\eta *}\\ C_{4}&{}-P+A_{4}X_{0}A_{4}^{\eta *}+C_{4}Y_{0}C_{4}^{\eta *}&{}0&{}A_{4}&{}0&{}0\\ D_{4}&{}0&{}Q-B_{4}Z_{0}B_{4}^{\eta *}-D_{4}Y_{0}D_{4}^{\eta *}&{}0&{}B_{4}&{}0\\ E_{2}&{}0&{}0&{}0&{}0&{}0\\ 0&{}0&{}0&{}E_{1}&{}0&{}0\\ 0&{}0&{}0&{}0&{}E_{3}&{}0 \end{bmatrix}\\&\quad =r\begin{bmatrix}C_{4}&{}A_{4}&{}0\\ D_{4}&{}0&{}B_{4}\\ E_{2}&{}0&{}0\\ 0&{}E_{1}&{}0\\ 0&{}0&{}E_{3}\end{bmatrix}+r\begin{bmatrix}C_{4}\\ D_{4}\\ E_{2}\end{bmatrix} \Leftrightarrow (3.18). \end{aligned}$$

Similarly, we can prove that $R_{A}C(R_{B})^{\eta *}=0$ $\Longleftrightarrow$ (3.19). $\square$

As a special case of system (3.1), we obtain a result about the system of quaternion matrix equations

$$\begin{aligned} \left\{ \begin{array}{c} A_{1}XA_{1}^{\eta *}+B_{1}YB_{1}^{\eta *}=E_{1},\\ A_{2}ZA_{2}^{\eta *}+B_{2}YB_{2}^{\eta *}=E_{2}, \end{array} \right. ~X=X^{\eta *},~Y=Y^{\eta *},~Z=Z^{\eta *}, \end{aligned}$$

(3.36)

which we state as follows:

Corollary 3.2

Let $A_{i},B_{i}$ and $E_{i}=E_{i}^{\eta *}$ be given over ${\mathbb {H}}$, $(\eta \in \{{\mathbf {i}},{\mathbf {j}},{\mathbf {k}}\},i=1,2)$ and

$$\begin{aligned} M_{i}=&\,R_{A_{i}}B_{i}, ~S_{i}=B_{i}L_{M_{i}},~ A_{3}=[L_{B_{1}},~L_{M_{1}}L_{S_{1}},~-L_{B_{2}},~-L_{M_{2}}L_{S_{2}}],\\ E_{3}=&\,\frac{1}{2}M_{2}^{\dag }E_{2}(B_{2}^{\dag })^{\eta *} [I+(S_{2}^{\dag }S_{2})^{\eta *}]+\frac{1}{2}(I+S_{2}^{\dag }S_{2})B_{2}^{\dag }E_{2}(M_{2}^{\dag })^{\eta *}\\&-\frac{1}{2}M_{1}^{\dag }E_{1}(B_{1}^{\dag })^{\eta *} [I+(S_{1}^{\dag }S_{1})^{\eta *}]-\frac{1}{2}(I+S_{1}^{\dag }S_{1})B_{1}^{\dag }E_{1}(M_{1}^{\dag })^{\eta *},\\ A=&\,R_{A_{3}}L_{M_{1}},~B=R_{A_{3}}L_{M_{2}},~C=R_{A_{3}}E_{3}(R_{A_{3}})^{\eta *}, ~ M=R_{A}B,~S=BL_{M}. \end{aligned}$$

Then, the following statements are equivalent:

• (1) System (3.36) has an $\eta$-Hermitian solution (X, Y, Z).

• (2)
  $$\begin{aligned} R_{M_{i}}R_{A_{i}}E_{i}=0, ~ R_{A_{i}}E_{i}L_{D_{i}}=0, ~ R_{M}R_{A}C=0,~R_{A}C(R_{B})^{\eta *}=0. \end{aligned}$$
• (3)
  $$\begin{aligned} r[E_{i},~A_{i},~B_{i}] =&\,r[A_{i},~B_{i}], ~ r\begin{bmatrix} E_{i} &{} A_{i}\\ B_{i}^{\eta *} &{} 0 \end{bmatrix} =r(A_{i})+r(B_{i}) ,\\ r\begin{bmatrix} 0&{}B_{1}^{\eta *}&{}B_{2}^{\eta *}&{}0&{}0\\ B_{1}&{}E_{1}&{}0&{}A_{1}&{}0\\ B_{2}&{}0&{}-E_{2}&{}0&{}A_{2} \end{bmatrix} =&\,r \begin{bmatrix} B_{1} &{} A_{1}&{}0\\ B_{2}&{}0&{}A_{2} \end{bmatrix}+r\begin{bmatrix}B_{1}\\ B_{2} \end{bmatrix},\\ r\begin{bmatrix} 0&{}B_{1}^{\eta *}&{}B_{2}^{\eta *}&{}0\\ B_{1}&{}-E_{1}&{}0&{}A_{1}\\ B_{2}&{}0&{}E_{2}&{}0\\ 0&{}0&{}A_{2}^{\eta *}&{}0 \end{bmatrix} =&\,r \begin{bmatrix} B_{1} &{} A_{1}\\ B_{2}&{}0 \end{bmatrix}+r\begin{bmatrix}B_{1}&{}0\\ B_{2}&{}A_{2}\end{bmatrix}. \end{aligned}$$

In this case, the general $\eta$-Hermitian solution to system (3.36) can be expressed as

$$\begin{aligned} X=&\,A_{1}^{\dag }E_{1}(A_{1}^{\dag })^{\eta *} -\frac{1}{2}A_{1}^{\dag }B_{1}M_{1}^{\dag }E_{1}[I+(B_{1}^{\dag })^{\eta *} S_{1}^{\eta *}](A_{1}^{\dag })^{\eta *}\\&-\frac{1}{2}A_{1}^{\dag }(I+S_{1}B_{1}^{\dag })E_{1}(M_{1}^{\dag })^{\eta *}B_{1}^{\eta *}(A_{1}^{\dag })^{\eta *}\\&-A_{1}^{\dag }S_{1}V_{1}S_{1}^{\eta *}(A_{1}^{\dag })^{\eta *} +L_{A_{1}}U_{1}+(L_{A_{1}}U_{1})^{\eta *},\\ Z=&\,A_{1}^{\dag }E_{2}(A_{2}^{\dag })^{\eta *}\\&-\frac{1}{2}A_{2}^{\dag } B_{2}M_{2}^{\dag }E_{2}[I+(B_{2}^{\dag })^{\eta *}S_{2}^{\eta *}](A_{2}^{\dag })^{\eta *} -\frac{1}{2}A_{2}^{\dag }(I+S_{2}B_{2}^{\dag })E_{2}(M_{2}^{\dag })^{\eta *}B_{2}^{\eta *}(A_{2}^{\dag })^{\eta *}\\&+A_{2}^{\dag }S_{2}V_{2}S_{2}^{\eta *}(A_{2}^{\dag })^{\eta *} +L_{A_{2}}U_{4}+(L_{A_{2}}U_{4})^{\eta *},\\ Y=&\,\frac{1}{2}M_{1}^{\dag }E_{1}(B_{1}^{\dag })^{\eta *} [I+(S_{1}^{\dag }S_{1})^{\eta *}]+\frac{1}{2}(I+S_{1}^{\dag }S_{1})B_{1}^{\dag }E_{1}(M_{1}^{\dag })^{\eta *} +L_{M_{1}}V_{1}(L_{M_{1}})^{\eta *}\\&+L_{B_{1}}U_{2}+(L_{B_{1}}U_{2})^{\eta *} +L_{M_{1}}L_{S_{1}}U_{3}+(L_{M_{1}}L_{S_{1}}U_{3})^{\eta *}, \end{aligned}$$

or

$$\begin{aligned} Y=&\,\frac{1}{2}M_{2}^{\dag }E_{2}(B_{2}^{\dag })^{\eta *} [I+(S_{2}^{\dag }S_{2})^{\eta *}]+\frac{1}{2}(I+S_{2}^{\dag } S_{2})B_{2}^{\dag }E_{2}(M_{2}^{\dag })^{\eta *}-L_{M_{2}}V_{2}(L_{M_{2}})^{\eta *}\\&+L_{B_{2}}U_{5}+(L_{B_{2}}U_{5})^{\eta *} +L_{M_{2}}L_{S_{2}}U_{6}+(L_{M_{2}}L_{S_{2}}U_{6})^{\eta *}, \end{aligned}$$

where

$$\begin{aligned} U_{2}=&\,[I_{m},~0,~0,~0]\Bigg [A_{3}^{\dag }[E_{3}-L_{M_{1}} V_{1}(L_{M_{1}})^{\eta *}-L_{M_{2}}V_{2}(L_{M_{2}})^{\eta *}]\\&+L_{A_{3}}W_{1}+W_{2}^{\eta *}(A_{3}^{\dag })^{\eta *}(A_{3})^{\eta *}\\&-\frac{1}{2}A_{3}^{\dag }[E_{3}-L_{M_{1}}V_{1}(L_{M_{1}})^{\eta *} -L_{M_{2}}V_{2}(L_{M_{2}})^{\eta *}](A_{3}^{\dag })^{\eta *}A_{3}^{\eta *} -A_{3}^{\dag }W_{2}A_{3}^{\eta *}\Bigg ],\\ U_{3}=&\,[0,~I_{m},~0,~0]\Bigg [A_{3}^{\dag }[E_{3}-L_{M_{1}}V_{1} (L_{M_{1}})^{\eta *}-L_{M_{2}}V_{2}(L_{M_{2}})^{\eta *}]\\&+L_{A_{3}}W_{1}+W_{2}^{\eta *}(A_{3}^{\dag })^{\eta *}(A_{3})^{\eta *}\\&-\frac{1}{2}A_{3}^{\dag }[E_{3}-L_{M_{1}}V_{1}(L_{M_{1}})^{\eta *} -L_{M_{2}}V_{2}(L_{M_{2}})^{\eta *}](A_{3}^{\dag })^{\eta *}A_{3}^{\eta *} -A_{3}^{\dag }W_{2}A_{3}^{\eta *}\Bigg ],\\ U_{5}=&\,[0,~0,~I_{m},~0]\Bigg [A_{3}^{\dag }[E_{3}-L_{M_{1}}V_{1} (L_{M_{1}})^{\eta *}-L_{M_{2}}V_{2}(L_{M_{2}})^{\eta *}]\\&+L_{A_{3}}W_{1}+W_{2}^{\eta *}(A_{3}^{\dag })^{\eta *}(A_{3})^{\eta *}\\&-\frac{1}{2}A_{3}^{\dag }[E_{3}-L_{M_{1}}V_{1}(L_{M_{1}})^{\eta *} -L_{M_{2}}V_{2}(L_{M_{2}})^{\eta *}](A_{3}^{\dag })^{\eta *}A_{3}^{\eta *} -A_{3}^{\dag }W_{2}A_{3}^{\eta *}\Bigg ],\\ U_{6}=&\,[0,~0,~0,~I_{m}]\Bigg [A_{3}^{\dag }[E_{3}-L_{M_{1}}V_{1} (L_{M_{1}})^{\eta *}-L_{M_{2}}V_{2}(L_{M_{2}})^{\eta *}]\\&+L_{A_{3}}W_{1}+W_{2}^{\eta *}(A_{3}^{\dag })^{\eta *}(A_{3})^{\eta *}\\&-\frac{1}{2}A_{3}^{\dag }[E_{3}-L_{M_{1}}V_{1}(L_{M_{1}})^{\eta *} -L_{M_{2}}V_{2}(L_{M_{2}})^{\eta *}](A_{3}^{\dag })^{\eta *}A_{3}^{\eta *} -A_{3}^{\dag }W_{2}A_{3}^{\eta *}\Bigg ],\\ V_{1}=&\,V_{1}^{\eta *}=A^{\dag }C(A^{\dag })^{\eta *} -\frac{1}{2}A^{\dag }BM^{\dag }C[I+(B^{\dag })^{\eta *}S^{\eta *}](A^{\dag })^{\eta *}\\&-\frac{1}{2}A^{\dag }(I+SB^{\dag })C(M^{\dag })^{\eta *}B^{\eta *}(A^{\dag })^{\eta *} \\&-A^{\dag }SW_{3}S^{\eta *}(A^{\dag })^{\eta *}+L_{A}W_{4}+(L_{A}W_{4})^{\eta *},\\ V_{2}=&\,V_{2}^{\eta *} = \frac{1}{2}M^{\dag }C(B^{\dag })^{\eta *}[I+(S^{\dag }S)^{\eta *}]+\frac{1}{2}(I+S^{\dag }S)B^{\dag }C(M^{\dag })^{\eta *}+L_{M}W_{3}(L_{M})^{\eta *}\\&+L_{B}W_{5}+(L_{B}W_{5})^{\eta *} +L_{M}L_{S}W_{6}+(L_{M}L_{S}W_{6})^{\eta *} , \end{aligned}$$

and $U_{1},U_{4},W_{1},W_{2},W_{3}=W_{3}^{\eta *},W_{4},W_{5},W_{6}$ are arbitrary matrices over ${\mathbb {H}}$ with appropriate sizes, m is the column number of $B_{1}$.

4 An Algorithm and an Example

In this section, we give an algorithm and an example to illustrate the main result. The algorithm for finding the $\eta$-Hermitian solution to the system of real quaternion matrix equations (3.1) is presented as follows.

Algorithm 4.1

The $\eta$-Hermitian solution to the system of quaternion matrix equations (3.1).

Input: The system of quaternion matrix equations (3.1).

• (i) Compute $E_{j},F_{j},(j=1,2,3)$ by (3.2).

• (ii) Compute $A_{5},B_{5},C_{5},D_{5},P_{1},Q_{1},M_{1},S_{1},M_{2},S_{2},A_{6},P_{2},A,B,C,M,S$ by (3.3)–(3.9).

• (iii) Compute the Moore–Penrose inverses conditions in Theorem 3.1 (2).

• (iv) Compute the rank conditions in Theorem 3.1 (3).

• (iv) If any of equations in Theorem 3.1 (2) or (3) fails, return

  “No $\eta$-Hermitian solution”.

• (v) Else compute the Moore–Penrose inverses in Theorem 3.1 (3.20)–(3.22).

Output: $\eta$-Hermitian solution (X, Y, Z) in Theorem 3.1.

Here is an example to illustrate Theorem 3.1.

Example 1

Given the matrices:

$$\begin{aligned} A_{1}=&\,\begin{bmatrix} {\mathbf {j}}&{}\quad {\mathbf {k}}&{}\quad 2+{\mathbf {j}}\\ {\mathbf {i}}&{}\quad 0&{}\quad 1\\ {\mathbf {i}}+{\mathbf {j}}&{}\quad {\mathbf {k}}&{}\quad 3+{\mathbf {j}} \end{bmatrix},~ A_{2}= \begin{bmatrix} {\mathbf {k}}&{}\quad 1&{}\quad 0\\ -{\mathbf {j}}&{}\quad {\mathbf {i}}&{}\quad 0\\ -{\mathbf {j}}+{\mathbf {k}}&{}\quad 1+{\mathbf {i}}&{}\quad 0 \end{bmatrix},~ A_{3}=\begin{bmatrix} {\mathbf {j}}-{\mathbf {k}}&{}\quad {\mathbf {i}}+2{\mathbf {j}}&{}\quad 1\\ {\mathbf {k}}&{}\quad -{\mathbf {j}}&{}\quad {\mathbf {i}}\\ {\mathbf {j}}&{}\quad {\mathbf {i}}+{\mathbf {j}}&{}\quad 1+{\mathbf {i}} \end{bmatrix},\\ B_{1}=&\, \begin{bmatrix} {\mathbf {i}}&{}\quad {\mathbf {j}}&{}\quad {\mathbf {k}}\\ -1&{}\quad {\mathbf {k}}&{}\quad -{\mathbf {j}}\\ -1+{\mathbf {i}}&{}\quad {\mathbf {j}}+{\mathbf {k}}&{}\quad -{\mathbf {j}}+{\mathbf {k}} \end{bmatrix},~ B_{2}=\begin{bmatrix} 1+2{\mathbf {j}}&{}\quad 1-{\mathbf {k}}&{}\quad 0\\ -2{\mathbf {j}}&{}\quad {\mathbf {k}}&{}\quad 1\\ 1&{}\quad 1&{}\quad 1 \end{bmatrix},\\ B_{3}=&\,\begin{bmatrix} {\mathbf {i}}+{\mathbf {j}}-{\mathbf {k}}&{}\quad {\mathbf {j}}&{}\quad {\mathbf {i}}\\ {\mathbf {k}}&{}\quad -{\mathbf {j}}&{}\quad 1\\ {\mathbf {i}}+{\mathbf {j}}&{}\quad 0&{}\quad 1+{\mathbf {i}} \end{bmatrix},\\ C_{1}=&\,\begin{bmatrix} 1+3{\mathbf {i}}-2{\mathbf {j}}+2{\mathbf {k}}&{}\quad -1+{\mathbf {j}} &{}\quad 3+{\mathbf {i}}+2{\mathbf {j}}\\ -2{\mathbf {j}}+{\mathbf {k}}&{}\quad {\mathbf {k}}&{}\quad 2-{\mathbf {j}}+{\mathbf {k}}\\ 1+3{\mathbf {i}}-4{\mathbf {j}}+3{\mathbf {k}}&{}\quad -1+{\mathbf {j}} +{\mathbf {k}}&{}\quad 5+{\mathbf {i}}+{\mathbf {j}}+{\mathbf {k}} \end{bmatrix},\\ C_{2}=&\,\begin{bmatrix} {\mathbf {i}}&{}\quad {\mathbf {j}}+2{\mathbf {k}}&{}\quad -{\mathbf {i}}+2{\mathbf {k}}\\ -1&{}\quad -2{\mathbf {j}}+{\mathbf {k}}&{}\quad 1-2{\mathbf {j}}\\ -1+{\mathbf {i}}&{}\quad -{\mathbf {j}}+3{\mathbf {k}}&{}\quad 1-{\mathbf {i}}-2{\mathbf {j}}+2{\mathbf {k}} \end{bmatrix},\\ C_{3}=&\,\begin{bmatrix} 4+2{\mathbf {i}}+{\mathbf {j}}-3{\mathbf {k}}&{}\quad -2+4{\mathbf {i}} -{\mathbf {k}}&{}\quad 1-{\mathbf {j}}-{\mathbf {k}}\\ -3&{}\quad -3{\mathbf {i}}+{\mathbf {k}}&{}\quad -{\mathbf {i}}+{\mathbf {j}}\\ 1+2{\mathbf {i}}+{\mathbf {j}}-3{\mathbf {k}}&{}\quad -2+{\mathbf {i}}&{}\quad -1-{\mathbf {i}}-{\mathbf {k}} \end{bmatrix},\\ D_{1}=&\, \begin{bmatrix} -2{\mathbf {k}}&{}\quad -2&{}\quad 2{\mathbf {i}}\\ -{\mathbf {i}}+{\mathbf {k}}&{}\quad 1-{\mathbf {j}}&{}\quad -{\mathbf {i}}-{\mathbf {k}}\\ -2+2{\mathbf {i}}+{\mathbf {j}}+{\mathbf {k}}&{}\quad 1-{\mathbf {i}} +2{\mathbf {j}}+2{\mathbf {k}}&{}\quad -1-{\mathbf {i}}-2{\mathbf {j}}+2{\mathbf {k}} \end{bmatrix},\\ D_{2}=&\,\begin{bmatrix} 2+{\mathbf {j}}-2{\mathbf {k}}&{}\quad 2&{}\quad 2+{\mathbf {i}}+{\mathbf {j}}\\ 5{\mathbf {i}}+2{\mathbf {k}}&{}\quad -2+{\mathbf {i}}+{\mathbf {j}}&{}\quad 2{\mathbf {k}}\\ 5+3{\mathbf {j}}&{}\quad 3+{\mathbf {i}}-{\mathbf {j}}+2{\mathbf {k}}&{}\quad 1 \end{bmatrix},~\\ D_{3}=&\,\begin{bmatrix} -2+2{\mathbf {i}}+{\mathbf {j}}+3{\mathbf {k}}&{}\quad 2-{\mathbf {j}} +{\mathbf {k}}&{}\quad -1+{\mathbf {i}}+2{\mathbf {j}}\\ 1+3{\mathbf {i}}+{\mathbf {j}}+{\mathbf {k}}&{}\quad 2+{\mathbf {i}} +{\mathbf {j}}&{}\quad {\mathbf {i}}+3{\mathbf {k}}\\ -1-{\mathbf {i}}+{\mathbf {k}}&{}{\mathbf {k}}&{}\quad -2-{\mathbf {i}} \end{bmatrix},\\ A_{4}=&\,\begin{bmatrix} {\mathbf {j}}&{}\quad {\mathbf {k}}&{}\quad 1\\ 0&{}\quad 1&{}\quad {\mathbf {i}}\\ {\mathbf {j}}&{}\quad 1+{\mathbf {k}}&{}\quad 1+{\mathbf {i}} \end{bmatrix},~ B_{4}=\begin{bmatrix} 1&{}\quad {\mathbf {i}}+2{\mathbf {k}}&{}\quad 0\\ {\mathbf {j}}&{}\quad {\mathbf {k}}&{}\quad {\mathbf {i}}\\ 1+{\mathbf {j}}&{}\quad {\mathbf {i}}+3{\mathbf {k}}&{}\quad {\mathbf {i}} \end{bmatrix},\\ C_{4}=&\,\begin{bmatrix} 2&{}\quad 1&{}\quad {\mathbf {k}}\\ {\mathbf {j}}&{}\quad 0&{}\quad 1\\ 2+{\mathbf {j}}&{}\quad 1&{}\quad 1+{\mathbf {k}} \end{bmatrix},~D_{4}= \begin{bmatrix} 2{\mathbf {i}}+{\mathbf {k}}&{}\quad {\mathbf {j}}&{}\quad 1\\ 0&{}\quad {\mathbf {i}}&{}\quad {\mathbf {k}}\\ 2{\mathbf {i}}+{\mathbf {k}}&{}\quad {\mathbf {i}}+{\mathbf {j}}&{}\quad 1+{\mathbf {k}} \end{bmatrix},\\ P=&\,P^{{\mathbf {j}}*}= \begin{bmatrix} -1+11{\mathbf {i}}+7{\mathbf {k}}&{}\quad 2+3{\mathbf {i}} +3{\mathbf {j}}-{\mathbf {k}}&{}\quad 1+14{\mathbf {i}}+3{\mathbf {j}}+6{\mathbf {k}}\\ 2+3{\mathbf {i}}-3{\mathbf {j}}-{\mathbf {k}}&{}\quad -3 +{\mathbf {i}}&{}\quad -1+4{\mathbf {i}}-3{\mathbf {j}}-{\mathbf {k}}\\ 1+14{\mathbf {i}}-3{\mathbf {j}}+6{\mathbf {k}}&{}\quad -1+4{\mathbf {i}} +3{\mathbf {j}}-{\mathbf {k}}&{}\quad 18{\mathbf {i}}+5{\mathbf {k}} \end{bmatrix},\\ Q=&\,Q^{{\mathbf {j}}*}= \begin{bmatrix} 3+13{\mathbf {i}}+3{\mathbf {k}}&{}\quad -7-{\mathbf {i}}-3{\mathbf {j}} -7{\mathbf {k}}&{}\quad -4+12{\mathbf {i}}-3{\mathbf {j}}-4{\mathbf {k}}\\ -7-{\mathbf {i}}+3{\mathbf {j}}-7{\mathbf {k}}&{}\quad {\mathbf {i}} -3{\mathbf {k}}&{}\quad -7+3{\mathbf {j}}-10{\mathbf {k}}\\ -4+12{\mathbf {i}}+3{\mathbf {j}}-4{\mathbf {k}} &{}\quad -7-3{\mathbf {j}}-10{\mathbf {k}}&{}\quad -11+12{\mathbf {i}}-14{\mathbf {k}} \end{bmatrix}. \end{aligned}$$

Now, we consider the ${\mathbf {j}}$-Hermitian solution to system (3.1), where $\eta ={\mathbf {j}}.$ Note that the rank of a quaternion matrix A can be quickly calculated by using the following property (see [42])

$$\begin{aligned} r(A)=\frac{1}{2}r\begin{bmatrix}A_{1}&{}A_{2}\\ -\overline{A_{2}}&{}\overline{A_{1}}\end{bmatrix}, \end{aligned}$$

where $A=A_{1}+A_{2}{\mathbf {j}},$ $A_{1}$ and $A_{2}$ are complex matrices, $\overline{A_{1}}$ means the conjugate of the matrix $A_{1}$. The Moore–Penrose inverse of a quaternion matrix A can be calculated in MATLAB by using SVD, i.e. $A^{\dag }=V^{*}\begin{bmatrix}D_{r}^{-1}&{}0\\ 0&{}0\end{bmatrix}U^{*},$ where U and V are unitary quaternion matrices, $D_{r}=\text{ diag }\{d_{1},\ldots ,d_{r}\}$ and the $d^{,}$s are the positive singular values of A. Note that

$$\begin{aligned}&r\begin{bmatrix}A_{j}&{}C_{j}\\ B_{1}^{{\mathbf {j}}*}&{}D_{1}^{{\mathbf {j}}*} \end{bmatrix}=r\begin{bmatrix}A_{j}\\ B_{1}^{{\mathbf {j}}*}\end{bmatrix} =3,~ \begin{bmatrix} A_{j}C_{j}^{{\mathbf {j}}*}&{}A_{j}D_{j}\\ B_{j}^{{\mathbf {j}}*}C_{j}^{{\mathbf {j}}*}&{}B_{j}^{{\mathbf {j}}*}D_{j} \end{bmatrix}= \begin{bmatrix} C_{j}A_{j}^{{\mathbf {j}}*}&{}C_{j}B_{j}\\ D_{j}^{{\mathbf {j}}*}A_{j}^{{\mathbf {j}}*}&{}D_{j}^{{\mathbf {j}}*}B_{j} \end{bmatrix},~(j=1,2,3),\\&r\begin{bmatrix} P&{}A_{4}&{}C_{4}\\ C_{1}A_{4}^{{\mathbf {j}}*}&{}A_{1}&{}0\\ D_{1}^{{\mathbf {j}}*}A_{4}^{{\mathbf {j}}*}&{}B_{1}^{{\mathbf {j}}*}&{}0\\ C_{2}C_{4}^{{\mathbf {j}}*}&{}0&{}A_{2}\\ D_{2}^{{\mathbf {j}}*}C_{4}^{{\mathbf {j}}*}&{}0&{}B_{2}^{{\mathbf {j}}*}\end{bmatrix}= r\begin{bmatrix} A_{4}&{}C_{4}\\ A_{1}&{}0\\ B_{1}^{{\mathbf {j}}*}&{}0\\ 0&{}A_{2}\\ 0&{}B_{2}^{{\mathbf {j}}*}\end{bmatrix}=6,\\&r\begin{bmatrix} P&{}A_{4}&{}C_{4}C_{2}^{{\mathbf {j}}*}&{}C_{4}D_{2}\\ C_{4}^{{\mathbf {j}}*}&{}0&{}A_{2}^{{\mathbf {j}}*}&{}B_{2}\\ C_{1}A_{4}^{{\mathbf {j}}*}&{}A_{1}&{}0&{}0\\ D_{1}^{{\mathbf {j}}*}A_{4}^{{\mathbf {j}}*}&{}B_{1}^{{\mathbf {j}}*}&{}0&{}0\end{bmatrix}= r\begin{bmatrix} A_{1}\\ B_{1}^{{\mathbf {j}}*}\\ A_{4}\end{bmatrix}+r\begin{bmatrix} A_{2}\\ B_{2}^{{\mathbf {j}}*}\\ C_{4}\end{bmatrix}=6,\\&r\begin{bmatrix} Q&{}B_{4}&{}D_{4}\\ C_{3}B_{4}^{{\mathbf {j}}*}&{}A_{3}&{}0\\ D_{3}^{{\mathbf {j}}*}B_{4}^{{\mathbf {j}}*}&{}B_{3}^{{\mathbf {j}}*}&{}0\\ C_{2}D_{4}^{{\mathbf {j}}*}&{}0&{}A_{2}\\ D_{2}^{{\mathbf {j}}*}D_{4}^{{\mathbf {j}}*}&{}0&{}B_{2}^{{\mathbf {j}}*}\end{bmatrix}= r\begin{bmatrix} B_{4}&{}D_{4}\\ A_{3}&{}0\\ B_{3}^{{\mathbf {j}}*}&{}0\\ 0&{}A_{2}\\ 0&{}B_{2}^{{\mathbf {j}}*}\end{bmatrix}=6,\\&r\begin{bmatrix} Q&{}B_{4}&{}D_{4}C_{2}^{{\mathbf {j}}*}&{}D_{4}D_{2}\\ D_{4}^{{\mathbf {j}}*}&{}0&{}A_{2}^{{\mathbf {j}}*}&{}B_{2}\\ C_{3}B_{4}^{{\mathbf {j}}*}&{}A_{3}&{}0&{}0\\ D_{3}^{{\mathbf {j}}*}B_{4}^{{\mathbf {j}}*}&{}B_{3}^{{\mathbf {j}}*}&{}0&{}0\end{bmatrix}= r\begin{bmatrix} A_{3}\\ B_{3}^{{\mathbf {j}}*}\\ B_{4}\end{bmatrix}+r\begin{bmatrix} A_{2}\\ B_{2}^{{\mathbf {j}}*}\\ D_{4}\end{bmatrix}=6,\\&r\begin{bmatrix} 0&{}-C_{4}^{{\mathbf {j}}*}&{}D_{4}^{{\mathbf {j}}*}&{}0&{}0&{}A_{2}^{{\mathbf {j}}*}&{}B_{2}\\ C_{4}&{}P&{}0&{}A_{4}&{}0&{}0&{}0\\ D_{4}&{}0&{}Q&{}0&{}B_{4}&{}D_{4}C_{2}^{{\mathbf {j}}*}&{}D_{4}D_{2}\\ A_{2}&{}C_{2}C_{4}^{{\mathbf {j}}*}&{}0&{}0&{}0&{}0&{}0\\ B_{2}^{{\mathbf {j}}*}&{}D_{2}^{{\mathbf {j}}*}C_{4}^{{\mathbf {j}}*}&{}0&{}0&{}0&{}0&{}0\\ 0&{}C_{1}A_{4}^{{\mathbf {j}}*}&{}0&{}A_{1}&{}0&{}0&{}0\\ 0&{}D_{1}^{{\mathbf {j}}*}A_{4}^{{\mathbf {j}}*}&{}0&{}B_{1}^{{\mathbf {j}}*}&{}0&{}0&{}0\\ 0&{}0&{}C_{3}B_{4}^{{\mathbf {j}}*}&{}0&{}A_{3}&{}0&{}0\\ 0&{}0&{}D_{3}^{{\mathbf {j}}*}B_{4}^{{\mathbf {j}}*}&{}0&{}B_{3}^{{\mathbf {j}}*}&{}0&{}0 \end{bmatrix}=r\begin{bmatrix}C_{4}\\ D_{4}\\ A_{2}\\ B_{2}^{{\mathbf {j}}*}\end{bmatrix}+ r\begin{bmatrix}C_{4}&{}A_{4}&{}0\\ D_{4}&{}0&{}B_{4}\\ A_{2}&{}0&{}0\\ B_{2}^{{\mathbf {j}}*}&{}0&{}0\\ 0&{}A_{1}&{}0\\ 0&{}B_{1}^{{\mathbf {j}}*}&{}0\\ 0&{}0&{}A_{3}\\ 0&{}0&{}B_{3}^{{\mathbf {j}}*}\end{bmatrix}=12, \end{aligned}$$

$$\begin{aligned}&r\begin{bmatrix} 0&{}-C_{4}^{{\mathbf {j}}*}&{}D_{4}^{{\mathbf {j}}*}&{}0&{}A_{2}^{{\mathbf {j}}*}&{}B_{2}&{}0&{}0\\ C_{4}&{}P&{}0&{}A_{4}&{}0&{}0&{}0&{}0\\ D_{4}&{}0&{}Q&{}0&{}D_{4}C_{2}^{{\mathbf {j}}*}&{}D_{4}D_{2}&{} B_{4}C_{3}^{{\mathbf {j}}*}&{}B_{4}D_{3}\\ A_{2}&{}C_{2}C_{4}^{{\mathbf {j}}*}&{}0&{}0&{}0&{}0&{}0&{}0\\ B_{2}^{{\mathbf {j}}*}&{}D_{2}^{{\mathbf {j}}*}C_{4}^{{\mathbf {j}}*}&{}0&{}0&{}0&{}0&{}0&{}0\\ 0&{}C_{1}A_{4}^{{\mathbf {j}}*}&{}0&{}A_{1}&{}0&{}0&{}0&{}0\\ 0&{}D_{1}^{{\mathbf {j}}*}A_{4}^{{\mathbf {j}}*}&{}0&{}B_{1}^{{\mathbf {j}}*}&{}0&{}0&{}0&{}0\\ 0&{}0&{}B_{4}^{{\mathbf {j}}*}&{}0&{}0&{}0&{}A_{3}^{{\mathbf {j}}*}&{}B_{3} \end{bmatrix}=r\begin{bmatrix}C_{4}&{}A_{4}\\ D_{4}&{}0\\ A_{2}&{}0\\ B_{2}^{{\mathbf {j}}*}&{}0\\ 0&{}A_{1}\\ 0&{}B_{1}^{{\mathbf {j}}*}\end{bmatrix}\\&\qquad +r\begin{bmatrix} C_{4}&{}0\\ D_{4}&{}B_{4}\\ A_{2}&{}0\\ B_{2}^{{\mathbf {j}}*}&{}0\\ 0&{}A_{3}\\ 0&{}B_{3}^{{\mathbf {j}}*}\end{bmatrix}=12. \end{aligned}$$

All equalities in (3.13)–(3.19) hold. By Theorem 3.1, we can conclude that system (3.1) has a ${\mathbf {j}}$-Hermitian solution (X, Y, Z),  where

$$\begin{aligned} X=&\,X^{{\mathbf {j}}*}= \begin{bmatrix}{\mathbf {k}}&{}{\mathbf {j}} &{}{\mathbf {j}}+{\mathbf {k}}\\ -{\mathbf {j}}&{}{\mathbf {i}}&{}0\\ -{\mathbf {j}}+{\mathbf {k}}&{}0&{}2\end{bmatrix},\\ Y=&\,Y^{{\mathbf {j}}*}=\begin{bmatrix}0&{}{\mathbf {i}}&{}2+{\mathbf {j}}\\ {\mathbf {i}}&{}2{\mathbf {k}}&{}0\\ 2-{\mathbf {j}}&{}0&{}1\end{bmatrix},\\ Z=&\,Z^{{\mathbf {j}}*}=\begin{bmatrix}{\mathbf {i}}&{}1+2{\mathbf {j}} &{}{\mathbf {i}}\\ 1-2{\mathbf {j}}&{}{\mathbf {k}}&{}0\\ {\mathbf {i}}&{}0&{}-1\end{bmatrix}. \end{aligned}$$

5 Conclusions

We have derived some practical necessary and sufficient conditions for the existence of an $\eta$-Hermitian solution (X, Y, Z) to the system of constrained two-sided coupled real quaternion matrix equations (1.3). We have also given the general $\eta$-Hermitian solution to system (1.3) when the solvability conditions are satisfied. An algorithm and a numerical example have been provided to illustrate the main result.