Some quaternion matrix equations involving $\phi$-Hermicity

Abstract

Let $\mathbb{H}$ be the real quaternion algebra and $\mathbb{H}^{m\times n}$ denote the set of all $m\times n$ matrices over $\mathbb{H}$. For $A\in\mathbb{H}^{m\times n},$ we denote by $A_{\phi}$ the $n\times m$ matrix obtained by applying $\phi$ entrywise to the transposed matrix $A^{t},$ where $\phi$ is a nonstandard involution of $\mathbb{H}$. $A\in\mathbb{H}^{n\times n}$ is said to be $\phi$-Hermitian if $A=A_{\phi}$. In this paper, we construct a  simultaneous decomposition of four real quaternion matrices with the same row number $(A,B,C,D),$ where $A$ is $\phi$-Hermitian, and $B,C,D$ are general matrices. Using  this simultaneous
matrix decomposition, we derive necessary and sufficient conditions for the existence
of a solution to some real quaternion matrix equations involving $\phi$-Hermicity in terms of
ranks of the given real quaternion matrices. We also present the general solutions to these real quaternion matrix equations when they are solvable. Finally some numerical examples are presented to illustrate the results of this paper.