A simultaneous decomposition of four real quaternion matrices encompassing $\eta$-Hermicity and its applications

TL;DR

This paper develops a simultaneous decomposition method for four real quaternion matrices, including $ ext{eta}$-Hermitian matrices, and applies it to solve related matrix equations with rank conditions.

Contribution

It introduces a novel simultaneous decomposition technique for four quaternion matrices with $ ext{eta}$-Hermicity, enabling solutions to complex matrix equations.

Findings

Derived necessary and sufficient conditions for quaternion matrix equations.

Provided general solutions for quaternion matrix equations involving $ ext{eta}$-Hermicity.

Illustrated results with numerical examples.

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 $\mathbb{H}$. Let $\mathbf{i},\mathbf{j},\mathbf{k}$ be the imaginary quaternion units. For $\eta\in\{\mathbf{i},\mathbf{j},\mathbf{k}\}$, a square real quaternion matrix $A$ is said to be $\eta$-Hermitian if $A^{\eta*}=A$ where $A^{\eta*}=-\eta A^{\ast}\eta$, and $A^{\ast}$ stands for the conjugate transpose of $A$. In this paper, we construct a simultaneous decomposition of four real quaternion matrices with the same row number $(A,B,C,D),$ where $A=A^{\eta*}\in \mathbb{H}^{m\times m}, B\in \mathbb{H}^{m\times p_{1}},C\in \mathbb{H}^{m\times p_{2}},D\in \mathbb{H}^{m\times p_{3}}$. As applications of this simultaneous matrix decomposition, we derive necessary and sufficient conditions for some real quaternion matrix equations involving $\eta$-Hermicity in terms of ranks of the coefficient matrices. We also present the general solutions to these real quaternion matrix equations. Moreover, we provide some numerical examples to illustrate our results.