Consistency and General Solutions to Some Sylvester-like Quaternion Matrix Equations

It is well known that quaternion and quaternion matrixes have a great range of applications in color image processing ([1,2]), quantum physics ([3,4]) and signal processing ([5]), etc. Quaternion matrix equations play an important role in mathematics and other domains, such as theoretical mechanics, optics, digital image processing, aerospace technology, etc. There are a great deal of papers from different aspects investigating quaternion matrix equation, such as solvability conditions, general solutions, extreme rank of solutions, minimum norm least squares solutions and their applications (e.g., [1,2,4,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31]).

Kyrchei investigated two-sided generalized Sylvester matrix equation $A_1{X}_1{B}_1+A_2{X}_2{B}_2=C$ over quaternion refer to Cramer’s Rules and Moore–Penrose inverse [13]. Liu et al. [17] considered the existence and uniqueness of solutions of some quaternion matrix equations with three new defined real representions. Kyrchei deduced the minimum norm least squares solutions of some quaternion matrix equations [11]. Futorny et al. [32] utilized Roth’s solvability criteria derived some solvability conditions for a few quaternion equations in terms of the equivalence of quaternion matrix. Xu et al. [28] provided some useful necessary and sufficient consitions and general solution to a constrained system of Sylvester-like matrix equations over the quaternion in terms of ranks and Moore–Penrose inverse of the coefficient matrixes. Mehany et al. [18] showed some solvaility conditions and general solution for three symmetrical coupled Sylvester-type matrix equations systems over quaternion from the aspects of ranks and generalized inverse. Liu et al. [15] gave solvability conditions and general solution to a Sylvester-like quaternion matrix equation $A_{1}X+Y{B}_1+A_2{Z}_1{B}_2+A_3{Z}_2{B}_3+A_4{Z}_3{B}_4=C$ with five unknown matrixes. Wang et al. [27] formed some practical solvability conditions for a Sylvester-like matrix equation system and gave an application involving an $\eta$-Hermitian solution. Dmytryshyn et al. [33] showed some solvability conditions for a class of quaternion matrix equations. He [10] gave some useful necessary and sufficient conditions for the existence of in terms of ranks and Moore–Penrose inverses. Wang et al. [25] considered a system of constrained two-sided coupled Sylvester-type quaternion matrix equations and gave some solvability conditions and general solution to the system. As far as we know, there has been little information on the solvabiliy conditions to the system in terms of quaternion matrix decomposition. One goal of this paper is to give some practical equivalent conditions for the consistency of the system (1) in terms of block matrixes and ranks.

$\varphi$-Hermitian quaternion matrix was put forward by Rodman in [34] at first, and some properties about it were given. In addition to general solutions, $\varphi$-Hermitian solutions of quaternion matrix equations play an important role in signal processing, engineering, systems and cybernetics, etc. With the need of practical application, the $\varphi$-Hermitian solutions of quaternion matrix equations have been paid more and more attention. He et al. [8] derived some practical necessary and sufficient conditions for the existence of the $\varphi$-Hermitian solution to the system of quaternion matrix equations where $A, B, C, D, \Omega$, and $\Theta$ are given quaternion matrixes. He et al. [7] considered the following systems of quaternion matrix equations involving $\varphi$-Hermicity and He [6] derived necessary and sufficient conditions for the existence of the solution to quaternion matrix equations and involving $\varphi$-herminity in terms ranks of the given quaternion matrixes. To the best of our knowledge, there is little information on the general $\varphi$-Hermitian solution to the system of quaternion matrix equations Another goal of this article is to give some solvability conditions and the expressions of the general $\varphi$-Hermitian solution to the system (2) via using the similar method with the system (1).

Let $\mathbb{R}$ stand for the real number field and $\mathbb{H}$ represent the real quaternion number field which is a four-dimensional linear space over $\mathbb{R}$ with a pattern that Let ${\mathbb{H}}^{m\times n}$ stand for all $m\times n$ matrixes over the real quaternion algebra, and if $A\in {\mathbb{H}}^{m\times n}$, then sign $A^T$ the transpose matrix of A. $A=A_{\varphi }$ is $\varphi$-Hermitian, if $A_{\varphi }$ is a $n\times m$ quaternion matrix obtained through applyling a nonstandard involution transform $\varphi$ with transformation matrix $\left(\begin{array}{cc}1& 0\\ 0& Q\end{array}\right)$ based on basis $\{1, \mathbf{i}, \mathbf{j}, \mathbf{k}\}$ on $A^T$, where Q is a $3\times 3$ real orthogonal symmetric matrix with eigenvalues $1, 1, -1$ [34]. The rank of a quaternion matrix A is defined to be the right linear independent columns maximum number of A and is represented by symbol $r\left(A\right)$. Note that A and $PAQ$ have the same rank for any invertible matrixes P and Q with appropriate sizes. The identity matrix and zero matrix with appropriate size are labeled by I and 0, respectively.

In this section, we deduce the consistency conditions of the system of quaternion matrix Equation (1), where $A, B, C, D, E, F, G, H, \Omega$ and $\Phi$ are given quaternion matrixes, through utilizing the equivalence canonical form of four matrixes.

According to the matrix product order principle, we can observe that coefficient matrixes A, B have the same number of columns, A, C, D have the same number of rows, E, F, G have the same number of columns, and E, H have the same number of rows, so they can be formed in the following two quaternion matrix arrays where $A\in {\mathbb{H}}^{m\times p}, B\in {\mathbb{H}}^{t\times p}, C\in {\mathbb{H}}^{m\times q}$, $D\in {\mathbb{H}}^{m\times s}$, $E\in {\mathbb{H}}^{p_1\times m_1}, H\in {\mathbb{H}}^{p_1\times t_1}, F\in {\mathbb{H}}^{q_1\times m_1}$ and $G\in {\mathbb{H}}^{s_1\times m_1}$.

To solve the equation system, we first give the simultaneous decomposition and equivalence canonical form of four quaternion matrixes in the following lemma.

We utilize Lemma 1 to transform the matrix arrays (3) into two simple forms such as (4) and where $S_a$, $S_b$, $S_c$, $S_d$, $S_e$, $S_f$, $S_g$, $S_h$ satisfy and and $M$, $P$, $T$, $Q$, $S$, $M_1$, $P_1$, $T_1$, $Q_1$, $S_1$ are nonsingular.

Hence, the system (1) is equivalent to

Set where ${\left({\Omega }_{ij}\right)}_{18\times 18}$ and ${\left({\Phi }_{ij}\right)}_{8\times 8}$ have the same block rows as $S_a$ and $S_b$, the same block columns as $S_e$ and $S_h$, respectively. Then, substituting (9)–(11) into the system (8) becomes

A result can be conclude that the system (1) is equivalent with the system (12).

Substituting (9)–(11) into the system (12) yields and where and