On Hermitian Solutions of the Generalized Quaternion Matrix Equation AXB+CX D=E

Abstract

The paper deals with the matrix equation $AXB+CXD=E$ over the generalized quaternions. By the tools of the real representation of a generalized quaternion matrix, Kronecker product as well as vec-operator, the paper derives the necessary and sufficient conditions for the existence of a Hermitian solution and gives the explicit general expression of the solution when it is solvable and provides a numerical example to test our results. The paper proposes a unificated algebraic technique for finding Hermitian solutions to the mentioned matrix equation over the generalized quaternions, which includes many important quaternion algebras, such as the Hamilton quaternions and the split quaternions.

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1. Introduction

In 1843, Irish Mathematician William Rowan Hamilton introduced the Hamilton quaternions. It is a great event in the history of mathematics. The set of Hamilton quaternions can form a skew field [1, 2]. In 1849, James Cockle introduced the split quaternions. It can be used to express Lorentzian rotations, which is used in geometry and physics (see [3–5]). In this paper, we consider a more generalized case, that is, the generalized quaternions, which is in the form of [6]$$\begin{array}{}\text{(1)}& q=q_1+q_{2}i+q_{3}j+q_{4}k,q_1,q_2,q_3,q_4\in \mathbf{R}.\end{array}$$where $i^2=u,j^2=v,k^2=ijk=-uv,ij=-ji=k,jk=-kj=-vi,ik=-ki=uj$. In the paper, we only focus on the cases of $0\ne u,v\in \mathbf{R}$. We call the set ${\mathbf{Q}}_G$ by the generalized quaternions. Obviously, ${\mathbf{Q}}_G$ is a noncommutative 4-dimensional Clifford algebra. Specially, ${\mathbf{Q}}_G$ is the Hamilton quaternion ring $\mathbf{Q}$ if $u=v=-1$, ${\mathbf{Q}}_G$ is the split quaternion ring ${\mathbf{Q}}_s$ if $u=-1,v=1$, ${\mathbf{Q}}_G$ is the nectarine ring ${\mathbf{Q}}_n$ if $u=1,v=-1$, ${\mathbf{Q}}_G$ is the conectarine ring ${\mathbf{Q}}_c$ if $u=v=1$.

Throughout this paper, let ${\mathbf{R}}^{m\times n}$, ${\mathbf{C}}^{m\times n}$, ${\mathbf{Q}}_G^{m\times n}$, $\mathbf{S}{\mathbf{R}}^{n\times n}$, and $\mathbf{AS}{\mathbf{R}}^{n\times n}$ denote the set of all $m\times n$ real matrices, the set of all $m\times n$ complex matrices, the set of all $m\times n$ generalized quaternion matrices, the set of all $n\times n$ real symmetric matrices, and the set of all $n\times n$ real antisymmetric matrices, respectively. The identity matrix of order $n$ is denoted by $I_n$. The zero matrix with suitable size is denoted by 0. We define the conjugate of $q\in {\mathbf{Q}}_G$ as $\overline{q}=q_1-q_{2}i-q_{3}j-q_{4}k$. For $A=a_{ij}\in {\mathbf{Q}}_G^{m\times n}$; we use $\overline{A}=\overline{a_{ij}}$, $A^T$ to denote the conjugate matrix, the transpose matrix of $A$, respectively. $A^H=A_1^T-A_2^T\mathbf{i}-A_3^T\mathbf{j}-A_4^T\mathbf{k}$ is the conjugate transpose of $A$. We call a matrix $A\in {\mathbf{Q}}_G^{n\times n}$ is Hermitian if $A^H=A$, which we denote it by $A\in H{Q}_G^{n\times n}$, where $H{Q}_G^{n\times n}$ is the set of all Hermitian generalized quaternion matrices with the size of $n\times n$.

In recent decades, different kinds of matrix equations over some quaternion algebras had been studied, such as the $A^{H}X+XA=B$, $AX=B$, $AXB=C$, $AXB+CYD=E$, and $AXB+CXD=E$, $A{X}^{\mathrm{\star }}-XB=CY+D$, and $X-A{X}^{\mathrm{\star }}B=C{Y}^{\mathrm{\star }}+D$ over the real/complex fields or some quaternion algebras (see [7–29] and references cited therein). For now, only few papers explored some fundamental properties and matrix equation over the generalized quaternions, which one may refer to [9, 17, 30, 31].

Hermitian matrix has attracted lots of attentions because of its great importance. There are some results about Hermitian solutions of matrix equations over several kinds of quaternion algebras (see [6, 26, 28, 32]). For example, Yu et al. [6] studied Hermitian solutions to the generalized quaternion matrix equation $AXB+C{X}^{\ast }D=E$ by the real representation method; Yuan et al. [32] discussed Hermitian solutions to the split quaternion matrix equation $AXB+CXD=E$ by using the complex representation method. Based on the work mentioned above, and inspired by the methods in ([28, 32]), we discuss the following problem:

Problem I: given $A\in {\mathbf{Q}}_G^{m\times n},B\in {\mathbf{Q}}_G^{n\times s}$, $C\in {\mathbf{Q}}_G^{m\times n},D\in {\mathbf{Q}}_G^{n\times s},\text{\hspace{0.17em}and\hspace{0.17em}}E\in {\mathbf{Q}}_G^{m\times s}$, find the solution set$$\begin{array}{}\text{(2)}& H_E=X|X\in \mathbf{H}{\mathbf{Q}}_G^{n\times n},AXB+CXD=E.\end{array}$$

2. Properties of the Generalized Quaternion Matrices

For any $A=A_1+A_2\mathbf{i}+A_3\mathbf{j}+A_4\mathbf{k}\in {\mathbf{Q}}_G^{m\times n}$ with $A_1,A_2,A_3,A_4\in {\mathbf{R}}^{m\times n}$, we define$$\begin{array}{}\text{(3)}& {\Phi }_A=A_1,A_2,A_3,A_4.\end{array}$$

Obviously, the map ${\mathrm{\Phi }}_A$ is an isomorphism of $A$, we denote by ${\mathrm{\Phi }}_A\cong A$. Next, we propose a real matrix representation for the generalized quaternion matrix $A\in {\mathbf{Q}}_G^{m\times n}$:$$\begin{array}{}\text{(4)}& A^R=\begin{array}{cccc}{A}_1& u{A}_2& v{A}_3& -uv{A}_4\\ A_2& A_1& v{A}_4& -v{A}_3\\ A_3& -u{A}_4& A_1& u{A}_2\\ A_4& -A_3& A_2& A_1\end{array}\in {\mathbf{R}}^{4m\times 4n}.\end{array}$$

For $A=a_{ij}\in {\mathbf{Q}}_G^{m\times n},B\in {\mathbf{Q}}_G^{p\times q}$, the Kronecker product of $A$ and $B$ is defined as $A\otimes B=a_{ij}B\in {\mathbf{Q}}_G^{mp\times nq}$. For the generalized quaternion matrices $A,B,C,D,E,F,G,\text{\hspace{0.17em}and\hspace{0.17em}}H$ with suitable sizes and the real number $k$, we have$$\begin{array}{c}\text{(5)}& kA\otimes C=A\otimes kC=kA\otimes C,\\ A,B,C\otimes D=A\otimes D,B\otimes D,C\otimes D,\\ \begin{array}{cc}E& G\\ F& H\end{array}\otimes K=\begin{array}{cc}E\otimes K& G\otimes K\\ F\otimes K& H\otimes K\end{array}.\end{array}$$

For the matrix $A=a_{ij}\in {\mathbf{Q}}_G^{m\times n}$, let $a_j=a_{1j},a_{2j},\dots ,a_{mj}$ with $j=\mathrm{1,2},\dots ,n$, we denote the vector $\text{vec}A$ by$$\begin{array}{}\text{(6)}& \text{vec}A={a_1,a_2,\dots ,a_n}^T.\end{array}$$

Throughout the paper, we denote$$\begin{array}{c}\text{(7)}& N_n=\begin{array}{cccc}{I}_n& 0& 0& 0\\ 0& u{I}_n& 0& 0\\ 0& 0& v{I}_n& 0\\ 0& 0& 0& -uv{I}_n\end{array},Q_n=\begin{array}{cccc}0& I_n& 0& 0\\ I_n& 0& 0& 0\\ 0& 0& 0& \frac{1}{v}{I}_n\\ 0& 0& -\frac{1}{v}{I}_n& 0\end{array},\\ L_n=\begin{array}{cccc}0& 0& I_n& 0\\ 0& 0& 0& -\frac{1}{u}{I}_n\\ I_n& 0& 0& 0\\ 0& \frac{1}{u}{I}_n& 0& 0\end{array},\\ S_n=\begin{array}{cccc}0& 0& 0& I_n\\ 0& 0& -I_n& 0\\ 0& I_n& 0& 0\\ I_n& 0& 0& 0\end{array}.\end{array}$$

The following are some properties of generalized quaternion matrices.

Proposition 1.

Let$q\in Q_G$,$A\in {\mathbf{Q}}_G^{m\times n}$,$B\in {\mathbf{Q}}_G^{n\times t}$, and$C\in {\mathbf{Q}}_G^{n\times n}$. Then,

(i) $q\overline{q}\ge 0$ does not hold in general

(ii) ${AB}^T\ne B^T{A}^T$ in general

(iii) $\overline{AB}\ne \overline{A}\overline{B}$ in general

(iv) ${A^H}^R\ne {A^R}^H$ in general

(v) ${\overline{C}}^{-1}\ne \overline{C^{-1}}$ in general

(vi) ${C^T}^{-1}\ne {C^{-1}}^T$ in general

Proof.

When $u=v=-1$ and $u=-1,v=1$, ${\mathbf{Q}}_G$ is the quaternion ring and the split quaternion ring, we can refer to [26, 32]), and the other cases can be easily obtained by direct calculation.

Some important properties of $A^R$ and ${\mathrm{\Phi }}_A$ are as follows.

Proposition 2.

Let $k\in \mathbf{R}$, $A,B\in {\mathbf{Q}}_G^{m\times n}$, $C\in {\mathbf{Q}}_G^{n\times t}$, and $D\in {\mathbf{Q}}_G^{n\times n}$. Then,

(i) $A=B$ if and only if $A^R=B^R$, $A=B$ if and only if ${\mathrm{\Phi }}_A={\mathrm{\Phi }}_B$

(ii) ${A+B}^R=A^R+B^R$, ${kA}^R=k{A}^R$, ${\mathrm{\Phi }}_{A+B}={\mathrm{\Phi }}_A+{\mathrm{\Phi }}_B$, ${\mathrm{\Phi }}_{kA}=k{\mathrm{\Phi }}_A$

(iii) ${\mathrm{\Phi }}_{AC}={\mathrm{\Phi }}_A{N}_n{C}^R{N}_t^{-1}$, ${AC}^R=A^R{C}^R$

(iv) If $D$ is invertible, then ${D^{-1}}^R={D^R}^{-1}$

(v) $I_n^R=I_{4n}$

Proof.

Since the proofs of (i), (ii), (iv), and (v) are easy, we only prove (iii). By direct calculation, we have$$\begin{array}{c}\text{(8)}& AC=A_1+A_2\mathbf{i}+A_3\mathbf{j}+A_4\mathbf{k}{C}_1+C_2\mathbf{i}+C_3\mathbf{j}+C_4\mathbf{k}=A_1{C}_1+u{A}_2{C}_2+v{A}_3{C}_3-uv{A}_4{C}_4+A_1{C}_2+A_2{C}_1-v{A}_3{C}_4+v{A}_4{C}_3\mathbf{i}\\ +A_1{C}_3+u{A}_2{C}_4+A_3{C}_1-u{A}_4{C}_2\mathbf{j}+A_1{C}_4+A_2{C}_3-A_3{C}_2+A_4{C}_1\mathbf{k}\\ =Z_1+Z_2\mathbf{i}+Z_3\mathbf{j}+Z_4\mathbf{k}.\end{array}$$

Thus,$$\begin{array}{}\text{(9)}& AC\cong {\Phi }_{AC}=\begin{array}{cccc}{Z}_1& Z_2& Z_3& Z_4\end{array},\end{array}$$where$$\begin{array}{c}\text{(10)}& Z_1=A_1{C}_1+u{A}_2{C}_2+v{A}_3{C}_3-uv{A}_4{C}_4,Z_2=A_1{C}_2+A_2{C}_1-v{A}_3{C}_4+v{A}_4{C}_3,\\ Z_3=A_1{C}_3+u{A}_2{C}_4+A_3{C}_1-u{A}_4{C}_2,\\ Z_4=A_1{C}_4+A_2{C}_3-A_3{C}_2+A_4{C}_1.\end{array}$$

Now, it is easy to verify $AC\cong {\mathrm{\Phi }}_{AC}=\begin{array}{cccc}{Z}_1& Z_2& Z_3& Z_4\end{array}={\mathrm{\Phi }}_A{N}_n{C}^R{N}_t^{-1}$.

3. The Structure of $\text{vec}AXB$

In the section, we investigate the structure of $\text{vec}AXB$. For $A\in {\mathbf{C}}^{m\times n}$, $B\in {\mathbf{C}}^{n\times s}$, and $C\in {\mathbf{C}}^{s\times t}$, it is well known that$$\begin{array}{}\text{(11)}& \text{vec}ABC=C^T\otimes A\text{vec}B.\end{array}$$

However, (11) cannot hold in the generalized quaternions for the noncommutative multiplication of the generalized quaternions. Thus, we need to study the structure of $\text{vec}{\mathrm{\Phi }}_{ABC}$.

Theorem 1.

Let $A=A_1+A_2\mathbf{i}+A_3\mathbf{j}+A_4\mathbf{k}\in {\mathbf{Q}}_G^{m\times n}$, $B=B_1+B_2\mathbf{i}+B_3\mathbf{j}+B_4\mathbf{k}\in {\mathbf{Q}}_G^{n\times s}$, and $C=C_1+C_2\mathbf{i}+C_3\mathbf{j}+C_4\mathbf{k}\in {\mathbf{Q}}_G^{s\times t}$, where $A_i\in {\mathbf{R}}^{m\times n},B_i\in {\mathbf{R}}^{n\times s}$, and $C_i\in {\mathbf{R}}^{s\times t}i=\mathrm{1,2,3,4}$. Then,$$\begin{array}{}\text{(12)}& \text{vec}{\Phi }_{ABC}={N_s{C}^R{N}_t^{-1}}^T\otimes A_1+{N_s{C}^R{Q}_t}^T\otimes A_2+{N_s{C}^R{L}_t}^T\otimes A_3+{N_s{C}^R{S}_t}^T\otimes A_4\text{vec}{\Phi }_B.\end{array}$$

Proof.

By (iii) in Proposition 2,$$\begin{array}{c}\text{(13)}& {\Phi }_{ABC}={\Phi }_A{N}_n{BC}^R{N}_t^{-1}={\Phi }_A{N}_n{B}^R{C}^R{N}_t^{-1}=\begin{array}{cccc}{A}_1& A_2& A_3& A_4\end{array}\begin{array}{cccc}{I}_n& 0& 0& 0\\ 0& u{I}_n& 0& 0\\ 0& 0& v{I}_n& 0\\ 0& 0& 0& -uv{I}_n\end{array}\begin{array}{cccc}{B}_1& u{B}_2& v{B}_3& -uv{B}_4\\ B_2& B_1& v{B}_4& -v{B}_3\\ B_3& -u{B}_4& B_1& u{B}_2\\ B_4& -B_3& B_2& B_1\end{array}\\ \times \begin{array}{cccc}{C}_1& u{C}_2& v{C}_3& -uv{C}_4\\ C_2& C_1& v{C}_4& -v{C}_3\\ C_3& -u{C}_4& C_1& u{C}_2\\ C_4& -C_3& C_2& C_1\end{array}\begin{array}{cccc}{I}_t& 0& 0& 0\\ 0& \frac{1}{u}{I}_t& 0& 0\\ 0& 0& \frac{1}{v}{I}_t& 0\\ 0& 0& 0& -\frac{1}{uv}{I}_t\end{array}\\ =\begin{array}{cccc}{\gamma }_1& {\gamma }_2& {\gamma }_3& {\gamma }_4\end{array},\end{array}$$where$$\begin{array}{c}\text{(14)}& {\gamma }_1=A_1{B}_1{C}_1+u{A}_2{B}_2{C}_1+v{A}_3{B}_3{C}_1-uv{A}_4{B}_4{C}_1+u{A}_1{B}_2{C}_2+u{A}_2{B}_1{C}_2-uv{A}_3{B}_4{C}_2+uv{A}_4{B}_3{C}_2+v{A}_1{B}_3{C}_3+uv{A}_2{B}_4{C}_3+v{A}_3{B}_1{C}_3-uv{A}_4{B}_2{C}_3-uv{A}_1{B}_4{C}_4-uv{A}_2{B}_3{C}_4+uv{A}_3{B}_2{C}_4-uv{A}_4{B}_1{C}_4,\\ {\gamma }_2=A_1{B}_1{C}_2+u{A}_2{B}_2{C}_2+v{A}_3{B}_3{C}_2-uv{A}_4{B}_4{C}_2+A_1{B}_2{C}_1+A_2{B}_1{C}_2-v{A}_3{B}_4{C}_1\\ +v{A}_4{B}_3{C}_1-v{A}_1{B}_3{C}_4-uv{A}_2{B}_4{C}_4-v{A}_3{B}_1{C}_4+uv{A}_4{B}_2{C}_4+v{A}_1{B}_4{C}_3+v{A}_2{B}_3{C}_3-v{A}_3{B}_2{C}_3+v{A}_4{B}_1{C}_3,\\ {\gamma }_3=A_1{B}_1{C}_3+u{A}_2{B}_2{C}_3+v{A}_3{B}_3{C}_3-uv{A}_4{B}_4{C}_3+u{A}_1{B}_2{C}_4+u{A}_2{B}_1{C}_4-uv{A}_3{B}_4{C}_4\\ +uv{A}_4{B}_3{C}_4+A_1{B}_3{C}_1+u{A}_2{B}_4{C}_1+A_3{B}_1{C}_1-u{A}_4{B}_2{C}_1-u{A}_1{B}_4{C}_2-u{A}_2{B}_3{C}_2+u{A}_3{B}_2{C}_2-u{A}_4{B}_1{C}_2,\\ {\gamma }_4=A_1{B}_1{C}_4+u{A}_2{B}_2{C}_4+v{A}_3{B}_3{C}_4-uv{A}_4{B}_4{C}_4+A_1{B}_2{C}_3+A_2{B}_1{C}_3-v{A}_3{B}_4{C}_3\\ +v{A}_4{B}_3{C}_3-A_1{B}_3{C}_2-u{A}_2{B}_4{C}_2-A_3{B}_1{C}_2+u{A}_4{B}_2{C}_2+A_1{B}_4{C}_1+A_2{B}_3{C}_1-A_3{B}_2{C}_1+A_4{B}_1{C}_1.\end{array}$$

It follows from (11) that$$\begin{array}{c}\text{(15)}& \text{vec}{\gamma }_1=C_1^T\otimes A_1\text{vec}{B}_1+u{C}_1^T\otimes A_2\text{vec}{B}_2+v{C}_1^T\otimes A_3\text{vec}{B}_3-uv{C}_1^T\otimes A_4\text{vec}{B}_4+u{C}_2^T\otimes A_1\text{vec}{B}_2+u{C}_2^T\otimes A_2\text{vec}{B}_1-uv{C}_2^T\otimes A_3\text{vec}{B}_4+uv{C}_2^T\otimes A_4\text{vec}{B}_3\\ +v{C}_3^T\otimes A_1\text{vec}{B}_3+uv{C}_3^T\otimes A_2\text{vec}{B}_4+v{C}_3^T\otimes A_3\text{vec}{B}_1-uv{C}_3^T\otimes A_4\text{vec}{B}_2\\ -uv{C}_4^T\otimes A_1\text{vec}{B}_4-uv{C}_4^T\otimes A_2\text{vec}{B}_3+uv{C}_4^T\otimes A_3\text{vec}{B}_2-uv{C}_4^T\otimes A_4\text{vec}{B}_1,\\ \text{vec}{\gamma }_2=C_2^T\otimes A_1\text{vec}{B}_1+u{C}_2^T\otimes A_2\text{vec}{B}_2+v{C}_2^T\otimes A_3\text{vec}{B}_3-uv{C}_2^T\otimes A_4\text{vec}{B}_4\\ +C_1^T\otimes A_1\text{vec}{B}_2+C_1^T\otimes A_2\text{vec}{B}_1-v{C}_1^T\otimes A_3\text{vec}{B}_4+v{C}_1^T\otimes A_4\text{vec}{B}_3\\ -v{C}_4^T\otimes A_1\text{vec}{B}_3-uv{C}_4^T\otimes A_2\text{vec}{B}_4-v{C}_4^T\otimes A_3\text{vec}{B}_1+uv{C}_4^T\otimes A_4\text{vec}{B}_2\\ +v{C}_3^T\otimes A_1\text{vec}{B}_4+v{C}_3^T\otimes A_2\text{vec}{B}_3-v{C}_3^T\otimes A_3\text{vec}{B}_2+v{C}_3^T\otimes A_4\text{vec}{B}_1,\\ \text{vec}{\gamma }_3=C_3^T\otimes A_1\text{vec}{B}_1+u{C}_3^T\otimes A_2\text{vec}{B}_2+v{C}_3^T\otimes A_3\text{vec}{B}_3-uv{C}_3^T\otimes A_4\text{vec}{B}_4\\ +u{C}_4^T\otimes A_1\text{vec}{B}_2+u{C}_4^T\otimes A_2\text{vec}{B}_1-uv{C}_4^T\otimes A_3\text{vec}{B}_4+uv{C}_4^T\otimes A_4\text{vec}{B}_3\\ +C_1^T\otimes A_1\text{vec}{B}_3+u{C}_1^T\otimes A_2\text{vec}{B}_4+C_1^T\otimes A_3\text{vec}{B}_1-u{C}_1^T\otimes A_4\text{vec}{B}_2\\ -u{C}_2^T\otimes A_1\text{vec}{B}_4-u{C}_2^T\otimes A_2\text{vec}{B}_3+u{C}_2^T\otimes A_3\text{vec}{B}_2-u{C}_2^T\otimes A_4\text{vec}{B}_1,\\ \text{vec}{\gamma }_4=C_4^T\otimes A_1\text{vec}{B}_1+u{C}_4^T\otimes A_2\text{vec}{B}_2+v{C}_4^T\otimes A_3\text{vec}{B}_3-uv{C}_4^T\otimes A_4\text{vec}{B}_4\\ +C_3^T\otimes A_1\text{vec}{B}_2+C_3^T\otimes A_2\text{vec}{B}_1-v{C}_3^T\otimes A_3\text{vec}{B}_4+v{C}_3^T\otimes A_4\text{vec}{B}_3\\ -C_2^T\otimes A_1\text{vec}{B}_3-u{C}_2^T\otimes A_2\text{vec}{B}_4-C_2^T\otimes A_3\text{vec}{B}_1+u{C}_2^T\otimes A_4\text{vec}{B}_2\\ +C_1^T\otimes A_1\text{vec}{B}_4+C_1^T\otimes A_2\text{vec}{B}_3-C_1^T\otimes A_3\text{vec}{B}_2+C_1^T\otimes A_4\text{vec}{B}_1.\end{array}$$

Thus,$$\begin{array}{c}\text{(16)}& \text{vec}{\Phi }_{ABC}=\begin{array}{c}\text{vec}{\gamma }_1\\ \text{vec}{\gamma }_2\\ \text{vec}{\gamma }_3\\ \text{vec}{\gamma }_4\end{array}={\begin{array}{cccc}{C}_1& C_2& C_3& C_4\\ u{C}_2& C_1& u{C}_4& C_3\\ v{C}_3& -v{C}_4& C_1& -C_2\\ -uv{C}_4& v{C}_3& -u{C}_2& C_1\end{array}}^T\otimes A_1\text{vec}{\Phi }_B\\ +{\begin{array}{cccc}u{C}_2& C_1& u{C}_4& C_3\\ u{C}_1& u{C}_2& u{C}_3& u{C}_4\\ -uv{C}_4& v{C}_3& -u{C}_2& C_1\\ uv{C}_3& -uv{C}_4& u{C}_1& -u{C}_2\end{array}}^T\otimes A_2\text{vec}{\Phi }_B\\ +{\begin{array}{cccc}v{C}_3& -v{C}_4& C_1& -C_2\\ uv{C}_4& -v{C}_3& u{C}_2& -C_1\\ v{C}_1& v{C}_2& v{C}_3& v{C}_4\\ -uv{C}_2& -v{C}_1& -uv{C}_4& -v{C}_3\end{array}}^T\otimes A_3\text{vec}{\Phi }_B\\ +{\begin{array}{cccc}-uv{C}_4& v{C}_3& -u{C}_2& C_1\\ -uv{C}_3& uv{C}_4& -u{C}_1& u{C}_2\\ uv{C}_2& v{C}_1& uv{C}_4& v{C}_3\\ -uv{C}_1& -uv{C}_2& -uv{C}_3& -uv{C}_4\end{array}}^T\otimes A_4\text{vec}{\Phi }_B\\ ={N_s{C}^R{N}_t^{-1}}^T\otimes A_1+{N_s{C}^R{Q}_t}^T\otimes A_2+{N_s{C}^R{L}_t}^T\otimes A_3+{N_s{C}^R{S}_t}^T\otimes A_4\text{vec}{\Phi }_B,\end{array}$$which completed our proof.

Yuan et al. [26] studied the $\text{vec}{\mathrm{\Phi }}_{ABC}$ over $\mathbf{Q}$, while Theorem 1 extends it to the result over ${\mathbf{Q}}_G$. As we can see that Theorem 1 maps the product of generalized quaternion matrices into the product of real matrices by using the real representation method, by this way, we can convert a generalized quaternion matrix equation into a real one.

In the following, we introduce some definitions and useful lemmas.

Definition 1.

For the matrix $A=a_{ij}\in {\mathbf{Q}}_G^{n\times n}$, let $a_1=a_{11},\sqrt{2}{a}_{21},\dots ,\sqrt{2}{a}_{n1}$, $a_2=a_{22},\sqrt{2}{a}_{32},\dots ,\sqrt{2}{a}_{n2},\dots$, $a_{n-1}=a_{n-1n-1},\sqrt{2}{a}_{nn-1}$, $a_n=a_{nn}$, we denote$$\begin{array}{}\text{(17)}& {\text{vec}}_{S}A={a_1,a_2,\dots ,a_{n-1},a_n}^T\in {\mathbf{Q}}_G^{nn+1/2}.\end{array}$$

Definition 2.

For the matrix $B=b_{ij}\in {\mathbf{Q}}_G^{n\times n}$, let $b_1=b_{21},b_{31},\dots ,b_{n1}$, $b_2=b_{32},b_{42},\dots ,b_{n2},\backslash \backslash \dots$, $b_{n-2}=b_{n-1n-2},b_{nn-2}$, $b_{n-1}=b_{nn-1}$, we denote$$\begin{array}{}\text{(18)}& {\text{vec}}_{A}B=\sqrt{2}{b_1,b_2,\dots ,b_{n-2},b_{n-1}}^T\in {\mathbf{Q}}_G^{nn-1/2}.\end{array}$$

Lemma 1 (see [27]).

Suppose $X\in {\mathbf{R}}^{n\times n}$, then$$\begin{array}{}\text{(19)}& i\text{If\hspace{0.17em}}X\in \mathbf{S}{\mathbf{R}}^{n\times n},\text{then\hspace{0.17em}vec}X=K_S{\text{vec}}_{S}X,\end{array}$$where ${\text{vec}}_{S}X$ is represented as (4), and the matrix $K_S\in {\mathbf{R}}^{n^2\times nn+1/2}$ is of the following form:$$\begin{array}{}\text{(20)}& K_S=\frac{1}{\sqrt{2}}\begin{array}{ccccccccccccc}\sqrt{2}{e}_1& e_2& \cdots & e_{n-1}& e_n& 0& 0& \cdots & 0& \cdots & 0& 0& 0\\ 0& e_1& \cdots & 0& 0& \sqrt{2}{e}_2& e_3& \cdots & e_n& \cdots & 0& 0& 0\\ 0& 0& \cdots & 0& 0& 0& e_2& \cdots & 0& \cdots & 0& 0& 0\\ ⋮& ⋮& & ⋮& ⋮& ⋮& ⋮& & ⋮& & ⋮& ⋮& ⋮\\ 0& 0& \cdots & e_1& 0& 0& 0& \cdots & 0& \cdots & \sqrt{2}{e}_{n-1}& e_n& 0\\ 0& 0& \cdots & 0& e_1& 0& 0& \cdots & e_2& \cdots & 0& e_{n-1}& \sqrt{2}{e}_n\end{array},\end{array}$$where $e_i$ is the $i$-th column of the identity matrix of order $n$.$$\begin{array}{}\text{(21)}& ii\text{If\hspace{0.17em}}X\in \mathbf{A}\mathbf{S}{\mathbf{R}}^{n\times n},\text{then\hspace{0.17em}vec}X=K_A{\text{vec}}_{A}X,\end{array}$$where ${\text{vec}}_{A}X$ is represented as (5), and the matrix $K_A\in {\mathbf{R}}^{n^2\times nn-1/2}$ is of the following form:$$\begin{array}{}\text{(22)}& K_A=\frac{1}{\sqrt{2}}\begin{array}{ccccccccccc}{e}_2& e_3& \cdots & e_{n-1}& e_n& 0& \cdots & 0& 0& \cdots & 0\\ -e_1& 0& \cdots & 0& 0& e_3& \cdots & e_{n-1}& e_n& \cdots & 0\\ 0& -e_1& \cdots & 0& 0& -e_2& \cdots & 0& 0& \cdots & 0\\ ⋮& ⋮& & ⋮& ⋮& ⋮& & ⋮& ⋮& & ⋮\\ 0& 0& \cdots & -e_1& 0& 0& \cdots & -e_2& 0& \cdots & e_n\\ 0& 0& \cdots & 0& -e_1& 0& \cdots & 0& -e_2& \cdots & -e_{n-1}\end{array},\end{array}$$where $e_i$ is the $i$-th column of $I_n$. Obviously, $K_S^T{K}_S=I_{nn+1/2}$, $K_A^T{K}_A=I_{nn-1/2}$.

By Lemma 1, we have the following.

Theorem 2.

For $X=X_1+X_2\mathbf{i}+X_3\mathbf{j}+X_4\mathbf{k}\in \mathbf{H}{\mathbf{Q}}_G^{n\times n}$, then$$\begin{array}{}\text{(23)}& \text{vec}{\Phi }_X=W\begin{array}{c}{\text{vec}}_S{X}_1\\ {\text{vec}}_A{X}_2\\ {\text{vec}}_A{X}_3\\ {\text{vec}}_A{X}_4\end{array},\end{array}$$in which$$\begin{array}{}\text{(24)}& W=\begin{array}{cccc}{K}_S& 0& 0& 0\\ 0& K_A& 0& 0\\ 0& 0& K_A& 0\\ 0& 0& 0& K_A\end{array}.\end{array}$$

Proof.

For any $X=X_1+X_2\mathbf{i}+X_3\mathbf{j}+X_4\mathbf{k}\in \mathbf{H}{\mathbf{Q}}_G^{n\times n},X_i\in {\mathbf{R}}^{n\times n}i=\mathrm{1,2,3,4}$, it is easy to see that$$\begin{array}{c}\text{(25)}& X\in \mathbf{H}{\mathbf{Q}}_G^{n\times n}⟺X_1^T=X_1,X_2^T=-X_2,\\ X_3^T=-X_3,\\ X_4^T=-X_4.\end{array}$$

Obviously, $X_1$ is symmetric, and $X_2$, $X_3$, and $X_4$ are antisymmetric. By Lemma 1, we have$$\begin{array}{c}\text{(26)}& \text{vec}{\Phi }_X=\begin{array}{c}\text{vec}{X}_1\\ \text{vec}{X}_2\\ \text{vec}{X}_3\\ \text{vec}{X}_4\end{array}=\begin{array}{c}{K}_S{\text{vec}}_S{X}_1\\ K_A{\text{vec}}_A{X}_2\\ K_A{\text{vec}}_A{X}_3\\ K_A{\text{vec}}_A{X}_4\end{array}\\ =\begin{array}{cccc}{K}_S& 0& 0& 0\\ 0& K_A& 0& 0\\ 0& 0& K_A& 0\\ 0& 0& 0& K_A\end{array}\begin{array}{c}{\text{vec}}_S{X}_1\\ {\text{vec}}_A{X}_2\\ {\text{vec}}_A{X}_3\\ {\text{vec}}_A{X}_4\end{array}\\ =W\begin{array}{c}{\text{vec}}_S{X}_1\\ {\text{vec}}_A{X}_2\\ {\text{vec}}_A{X}_3\\ {\text{vec}}_A{X}_4\end{array}.\end{array}$$

Combining Theorems 1 and 2, we yield the following result.

Theorem 3.

Let $A=A_1+A_2\mathbf{i}+A_3\mathbf{j}+A_4\mathbf{k}\in {\mathbf{Q}}_G^{m\times n}$, $X=X_1+X_2\mathbf{i}+X_3\mathbf{j}+X_4\mathbf{k}\in \mathbf{H}{\mathbf{Q}}_G^{n\times n}$, and $B=B_1+B_2\mathbf{i}+B_3\mathbf{j}+B_4\mathbf{k}\in {\mathbf{Q}}_G^{n\times s}$, where $A_i\in {\mathbf{R}}^{m\times n},X_i\in {\mathbf{R}}^{n\times n}$, and $B_i\in {\mathbf{R}}^{n\times s}i=\mathrm{1,2,3,4}$. Then,$$\begin{array}{}\text{(27)}& \text{vec}{\Phi }_{AXB}={N_n{B}^R{N}_s^{-1}}^T\otimes A_1+{N_n{B}^R{Q}_s}^T\otimes A_2+{N_n{B}^R{L}_s}^T\otimes A_3+{N_n{B}^R{S}_s}^T\otimes A_{4}W\begin{array}{c}{\text{vec}}_S{X}_3\\ {\text{vec}}_A{X}_3\\ {\text{vec}}_A{X}_3\\ {\text{vec}}_A{X}_3\end{array}.\end{array}$$

4. The Hermitian Solutions

Based on our earlier discussion, we now pay our attention to Problem I. The following notation is necessary for deriving a solution to Problem I. Let $A=A_1+A_2\mathbf{i}+A_3\mathbf{j}+A_4\mathbf{k}\in {\mathbf{Q}}_G^{m\times n}$, $B=B_1+B_2\mathbf{i}+B_3\mathbf{j}+B_4\mathbf{k}\in {\mathbf{Q}}_G^{n\times s}$, $C=C_1+C_2\mathbf{i}+C_3\mathbf{j}+C_4\mathbf{k}\in {\mathbf{Q}}_G^{m\times n}$, $D=D_1+D_2\mathbf{i}+D_3\mathbf{j}+D_4\mathbf{k}\in {\mathbf{Q}}_G^{n\times s}$, and $E=E_1+E_2\mathbf{i}+E_3\mathbf{j}+E_4\mathbf{k}\in {\mathbf{Q}}_G^{m\times s}$. In the remaining of the paper, we set$$\begin{array}{c}\text{(28)}& P={N_n{B}^R{N}_s^{-1}}^T\otimes A_1+{N_n{B}^R{Q}_s}^T\otimes A_2+{N_n{B}^R{L}_s}^T\otimes A_3+{N_n{B}^R{S}_s}^T\otimes A_{4}W+{N_n{D}^R{N}_s^{-1}}^T\otimes C_1+{N_n{D}^R{Q}_s}^T\otimes C_2+{N_n{D}^R{L}_s}^T\otimes C_3+{N_n{D}^R{S}_s}^T\otimes C_{4}W,\\ e=\begin{array}{c}\text{vec}{E}_1\\ \text{vec}{E}_2\\ \text{vec}{E}_3\\ \text{vec}{E}_4\end{array}.\end{array}$$

We also need the following lemma.

Lemma 2 (see [33]).

The matrix equation $Ax=b$, with $A\in {\mathbf{R}}^{m\times n}$ and $b\in {\mathbf{R}}^m$, has a solution $x\in {\mathbf{R}}^n$ if and only if$$\begin{array}{}\text{(29)}& A{A}^{+}b=b,\end{array}$$where $A^{+}$ is the Moore–Penrose inverse of the matrix $A$. In this case, it has the general solution$$\begin{array}{}\text{(30)}& x=A^{+}b+I_n-A^{+}Ay,\end{array}$$where $y\in {\mathbf{R}}^n$ is an arbitrary vector, and it has the unique solution $x=A^{+}b$ for the case when $\text{rank}A=n$. The solution of the matrix equation $Ax=b$ with the least norm is $x=A^{+}b$.

Theorem 4.

Let $A\in {\mathbf{Q}}_G^{m\times n},B\in {\mathbf{Q}}_G^{n\times s}$, $C\in {\mathbf{Q}}_G^{m\times n},D\in {\mathbf{Q}}_G^{n\times s}$, and $E\in {\mathbf{Q}}_G^{m\times s}$. Then, Problem I has a solution $X\in H_E$ if and only if$$\begin{array}{}\text{(31)}& P{P}^{+}e=e.\end{array}$$

If this condition satisfies, then$$\begin{array}{}\text{(32)}& H_E=X|\text{vec}{\Phi }_X=W{P}^{+}e+I_{2n^2-n}-P^{+}Py,\end{array}$$where $y\in {\mathbf{R}}^{2n^2-n}$ is an arbitrary vector.

Furthermore, if (31) holds, then the generalized quaternion matrix equation (2) has a unique solution $X\in H_E$ if and only if$$\begin{array}{}\text{(33)}& \text{rank}P=2n^2-n.\end{array}$$

In this case,$$\begin{array}{}\text{(34)}& H_E=X|\text{vec}{\Phi }_X=W{P}^{+}e.\end{array}$$

Proof.

By (ii) in Proposition 2 and Theorem 3, we have$$\begin{array}{c}\text{(35)}& AXB+CXD=E⟺{\Phi }_{AXB}+{\Phi }_{CXD}={\Phi }_E⟺\text{vec}{\Phi }_{AXB}+\text{vec}{\Phi }_{CXD}\\ =\text{vec}{\Phi }_E⟺P\begin{array}{c}{\text{vec}}_S{X}_1\\ {\text{vec}}_A{X}_2\\ {\text{vec}}_A{X}_3\\ {\text{vec}}_A{X}_4\end{array}=e.\end{array}$$

By Lemma 2, Problem I has a solution $X\in H_E$ if and only if (31) holds. If this condition satisfies, then$$\begin{array}{}\text{(36)}& \begin{array}{c}{\text{vec}}_S{X}_1\\ {\text{vec}}_A{X}_2\\ {\text{vec}}_A{X}_3\\ {\text{vec}}_A{X}_4\end{array}=P^{+}e+I_{2n^2-n}-P^{+}Py.\end{array}$$

Also by (23),$$\begin{array}{c}\text{(37)}& \text{vec}{\Phi }_X=W\begin{array}{c}{\text{vec}}_S{X}_1\\ {\text{vec}}_A{X}_2\\ {\text{vec}}_A{X}_3\\ {\text{vec}}_A{X}_4\end{array}=W{P}^{+}e+I_{2n^2-n}-P^{+}Py,\end{array}$$where $y\in {\mathbf{R}}^{2n^2-n}$ is an arbitrary vector. We can draw the conclusion (32). Furthermore, if (31) holds, Problem I has a unique solution $X\in H_E$ if and only if$$\begin{array}{}\text{(38)}& P^{+}P=I_{2n^2-n}.\end{array}$$

That is, (33) holds. In the case, we obtain (34).

5. Example

In this section, we give two examples to illustrate our results.

Example 1.

Consider the Hamilton quaternion matrix equation $AXB+CXD=E$, where$$\begin{array}{c}\text{(39)}& A=\begin{array}{cc}i& 1+j\end{array},B=\begin{array}{c}-2+i\\ j-2k\end{array},\\ C=\begin{array}{cc}k& 1+j\end{array},\\ D=\begin{array}{c}3k\\ 1-2i+j\end{array},\\ E=1+2i+3j-k.\end{array}$$

Obviously, the Hamilton quaternions mean $u=v=-1$. By (4) and (28), we easily get$$\begin{array}{c}\text{(40)}& B^R=\begin{array}{cccc}-2& -1& 0& 0\\ 0& 0& -1& 2\\ 1& -2& 0& 0\\ 0& 0& 2& 1\\ 0& 0& -2& -1\\ 1& -2& 0& 0\\ 0& 0& 1& -2\\ -2& -1& 0& 0\end{array},D^R=\begin{array}{cccc}0& 0& 0& -3\\ 1& 2& -1& 0\\ 0& 0& -3& 0\\ -2& 1& 0& 1\\ 0& 3& 0& 0\\ 1& 0& 1& 2\\ 3& 0& 0& 0\\ 0& -1& -2& 1\end{array},\\ P={N_2{B}^R{N}_1^{-1}}^T\otimes A_1+{N_2{B}^R{Q}_1}^T\otimes A_2+{N_2{B}^R{L}_1}^T\otimes A_3+{N_2{B}^R{S}_1}^T\otimes A_{4}W\\ +{N_2{D}^R{N}_1^{-1}}^T\otimes C_1+{N_2{D}^R{Q}_1}^T\otimes C_2+{N_2{D}^R{L}_1}^T\otimes C_3+{N_2{D}^R{S}_1}^T\otimes C_{4}W,\\ e=\begin{array}{c}1\\ 2\\ 3\\ -1\end{array}.\end{array}$$

By Theorem 4 and MATLAB, calculating the formula $\text{vec}{\mathrm{\Phi }}_X=W{P}^{+}e$ gives a solution$$\begin{array}{}\text{(41)}& X=\begin{array}{cc}-0.0778& -0.3899\\ -0.3899& 0.1907\end{array}+\begin{array}{cc}0& 0.5091\\ -0.5091& 0\end{array}i+\begin{array}{cc}0& -0.3180\\ 0.3180& 0\end{array}j+\begin{array}{cc}0& 0.2478\\ -0.2478& 0\end{array}k.\end{array}$$

Example 2.

Consider the generalized quaternion matrix equation $AXB+CXD=E$ with $u=2,v=3$, where$$\begin{array}{c}\text{(42)}& A=\begin{array}{cc}j& 1-i\end{array},B=\begin{array}{c}-1-i\\ j+2k\end{array},\\ C=\begin{array}{cc}-k& 1-k+2j\end{array},\\ D=\begin{array}{c}-k\\ 1-j\end{array},\\ E=1-2i+j-3k.\end{array}$$

By (4) and (28), we easily get$$\begin{array}{c}\text{(43)}& B^R=\begin{array}{cccc}-1& -2& 0& 0\\ 0& 0& 3& -12\\ -1& -1& 0& 0\\ 0& 0& 6& -3\\ 0& 0& -1& -2\\ 1& -4& 0& 0\\ 0& 0& -1& -1\\ 2& -1& 0& 0\end{array},D^R=\begin{array}{cccc}0& 0& 0& 6\\ 1& 0& -3& 0\\ 0& 0& -3& 0\\ 0& 1& 0& 3\\ 0& 2& 0& 0\\ -1& 0& 1& 0\\ -1& 0& 0& 0\\ 0& 1& 0& 1\end{array},\\ P={N_2{B}^R{N}_1^{-1}}^T\otimes A_1+{N_2{B}^R{Q}_1}^T\otimes A_2+{N_2{B}^R{L}_1}^T\otimes A_3+{N_2{B}^R{S}_1}^T\otimes A_{4}W\\ +{N_2{D}^R{N}_1^{-1}}^T\otimes C_1+{N_2{D}^R{Q}_1}^T\otimes C_2+{N_2{D}^R{L}_1}^T\otimes C_3+{N_2{D}^R{S}_1}^T\otimes C_{4}W,\\ e=\begin{array}{c}1\\ -2\\ 1\\ -3\end{array}.\end{array}$$

By Theorem 4 and MATLAB, calculating the formula $\text{vec}{\mathrm{\Phi }}_X=W{P}^{+}e$ gives a solution$$\begin{array}{}\text{(44)}& X=\begin{array}{cc}0.8970& -0.1826\\ -0.1826& -0.2064\end{array}+\begin{array}{cc}0& 0.2769\\ -0.2769& 0\end{array}i+\begin{array}{cc}0& -0.3228\\ 0.3228& 0\end{array}j+\begin{array}{cc}0& -0.0111\\ 0.0111& 0\end{array}k.\end{array}$$

6. Conclusion

In this paper, we provide a direct method to find Hermitian solutions of the generalized quaternion matrix equation $AXB+CXD=E$ by using the real representation of generalized quaternion matrices, Kronecker product and vec-operator. We give the necessary and sufficient conditions for the existence of a Hermitian solution and also derive the general solution when the matrix equation is consistent. The paper proposes an algebraic technique for finding the Hermitian solutions to the above matrix equation over the generalized quaternions. The generalized quaternions include many important quaternion algebras, for instance, $\mathbf{Q}$, ${\mathbf{Q}}_s$, ${\mathbf{Q}}_n$, and ${\mathbf{Q}}_c$, thus the paper actually proposes a unified technique to solve the Hermitian solution problems over the several quaternion algebras.

Acknowledgments

This research was supported by Macao Science and Technology Development Fund (No. 185/2 017/A3) and The Joint Research and Development Fund of Wuyi University, Hong Kong and Macao (2019WGALH20).

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