Solution to a system of real quaternion matrix equations encompassing η-Hermicity

Abstract

Let ${\mathbb{H}}^{m\times n}$ be the set of all m × n matrices over the real quaternion algebra $\mathbb{H}=\{c_0+c_{1}i+c_{2}j+c_{3}k\mid i^2=j^2=k^2=ijk=-1,c_0,c_1,c_2,c_3\in \mathbb{R}\}$. $A\in {\mathbb{H}}^{n\times n}$ is known to be η-Hermitian if $A=A^{\eta *}=-\eta A^{*}\eta ,\eta \in \{i,j,k\}$ and A^* means the conjugate transpose of A. We mention some necessary and sufficient conditions for the existence of the solution to the system of real quaternion matrix equations including η-Hermicity

$$\begin{array}{ccc}\hfill A_{1}X& =& C_1,\hfill \\ \hfill A_{2}Y& =& C_2,Y{B}_2=D_2,Y=Y^{\eta *},\hfill \\ \hfill A_{3}Z& =& C_3,Z{B}_3=D_3,Z=Z^{\eta *},\hfill \\ \hfill A_{4}X& +& {\left(A_{4}X\right)}^{\eta *}+B_{4}Y{B}_4^{\eta *}+C_{4}Z{C}_4^{\eta *}=D_4,\hfill \end{array}$$

and also construct the general solution to the system when it is consistent. The outcome of this paper diversifies some particular results in the literature. Furthermore, we constitute an algorithm and a numerical example to comprehend the approach established in this treatise.

Introduction

In the whole exposition, we symbolize the real number field by $\mathbb{R},$ the complex number field by $\mathbb{C}$ and $\mathbb{H}$ for the quaternion algebra

$$\mathbb{H}=\{b_0+b_{1}i+b_{2}j+b_{3}k\mid i^2=j^2=k^2=ijk=-1,b_0,b_1,b_2,b_3\in \mathbb{R}\}.$$

The set of all matrices of dimension m × n over $\mathbb{H}$ is represented by ${\mathbb{H}}^{m\times n}$. I denotes an identity matrix with right size. For any matrix A over $\mathbb{H},$ the column right space and the row left space of A are denoted by $\mathcal{R}\left(A\right)$ and $\mathcal{N}\left(A\right),$ respectively. $D\left[\mathcal{R}\right(A\left)\right]$ denotes the dimension of $\mathcal{R}\left(A\right)$. By Hungerford [30], we have $D\left[\mathcal{R}\right(A\left)\right]=D\left[\mathcal{N}\right(A\left)\right],$ which is known as rank of A. A^* means the conjugate transpose of A. A^† means the Moore–Penrose inverse of $A\in {\mathbb{H}}^{m\times n},$ i.e., the exclusive matrix $Y\in {\mathbb{H}}^{n\times m}$ obeying

$$AYA=A,YAY=Y,{\left(AY\right)}^{*}=AY,{\left(YA\right)}^{*}=YA.$$

For significant findings on generalized inverses, one can see [1] and [7]. Furthermore, $L_A=I-A^{†}A$ and $R_A=I-A{A}^{†}$ denote the projectors generated by A, respectively. By the definition of the Moore–Penrose inverse, these projectors are Hermitian and idempotent as well.

The quaternions were first explored by the Irish mathematician Sir William Rowan Hamilton in [2]. Quaternions have massive applications in diverse areas of mathematics like computation, geometry and algebra; see, e.g. [3], [4], [5], [6]. Shoemake [8] introduced them in the field of computer graphics. Nowadays quaternion matrices play a remarkable role in control theory, mechanics, altitude control, quantum physics and signal processing; see, e.g. [9], [10]. In skeletal animation systems, quaternions are mostly applied to interpolate between joint orientations specified with key frames or animation curves [11]. For comprehensive study of quaternions, we refer to [12].

The Hermitian solution to some matrix equations were inspected by many writers in various papers. For instance, Khatri and Mitra [29] furnished some necessary and sufficient conditions for the presence of the Hermitian solution to

$$\begin{array}{ccc}\hfill AY& =& B,\hfill \\ \hfill A_{2}Y& =& C_2,Y{B}_2=D_2,\hfill \end{array}$$

over $\mathbb{C}$ respectively, and also presented a general Hermitian solutions to (1.1) with the technique of generalized inverses. The Hermitian solution to

$$\begin{array}{c}\hfill AX{A}^{*}=B\end{array}$$

was considered by Groß in [31].

Linear matrix equations have been one of the principal topics in matrix theory and its applications; see, e.g. [13], [14], [15], [16], [17], [18], [26], [35], [36], [37], [38], [39], [40], [41], [42], [43], [44], [45], [46]. The frequently studied Lyapunov equation

$$\begin{array}{c}\hfill BX+{\left(BX\right)}^{*}=A\end{array}$$

constitutes a major function in mathematics, such as optimal control, stability analysis, system theory, model reduction; see, e.g, [19], [20]. The classical matrix equation

$$\begin{array}{c}\hfill CX{C}^{*}+DY{D}^{*}=A\end{array}$$

has been analyzed by numerous authors with different approaches; see, e.g. [21], [22], [23]. In [24], Yuan and Wang derived the expression of the least squares η-Hermitian solution to the real quaternion matrix expression $AXB+CXD=E$.

He and Wang [32] gave the general solution of

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

having η-Hermicity over $\mathbb{H}$. Furthermore, Zhang and Wang [28] obtained the η-Hermitian solution to

$$\begin{array}{ccc}& & A_{1}X=C_1,\hfill \\ & & A_{2}Y=C_2,Z{B}_2=D_2,\hfill \\ & & A_{4}X+{\left(A_{4}X\right)}^{\eta *}+B_{4}Y{B}_4^{\eta *}+C_{4}Z{C}_4^{\eta *}=D_4.\hfill \end{array}$$

Very recently, He and Wang [33] derived the η-Hermitian solution to the system

$$\begin{array}{ccc}& & A_{1}Y=C_1,Y{B}_1=D_1,\hfill \\ & & A_{2}Z=C_2,Z{B}_2=D_2,\hfill \\ & & C_{3}Y{C}_3^{\eta *}+D_{3}Z{D}_3^{\eta *}=A_3.\hfill \end{array}$$

Observe that the system (1.2)–(1.4) are particular cases of the following system of quaternion matrix equations

$$\begin{array}{ccc}& & A_{1}X=C_1,\hfill \\ & & A_{2}Y=C_2,Y{B}_2=D_2,Y=Y^{\eta *},\hfill \\ & & A_{3}Z=C_3,Z{B}_3=D_3,Z=Z^{\eta *},\hfill \\ & & A_{4}X+{\left(A_{4}X\right)}^{\eta *}+B_{4}Y{B}_4^{\eta *}+C_{4}Z{C}_4^{\eta *}=D_4.\hfill \end{array}$$

To our best knowledge, there has been little research on the general solution of (1.5). Motivated by the above said findings, we in this paper establish the solvability conditions and the expression of the general solution to the solvable system (1.5).

The rest of this paper is composed as follows. In Section 2, we start with renowned results which are frequently experienced in the sequel. In Section 3, we offer some necessary and sufficient conditions for the presence of a solution (X, Y, Z) to (1.5), where Y and Z are η-Hermitian and present its general solution when the necessary and sufficient conditions are fulfilled. In Section 4, we discuss some particular cases of system (1.5). In Section 5, we present an algorithm and a numerical example to exemplify the notion established in this treatise. In Section 6, we give a conclusion to this paper.