A simultaneous decomposition of seven matrices over real quaternion algebra and its applications

TL;DR

This paper develops a simultaneous decomposition method for seven matrices over real quaternion algebra and applies it to analyze solvability, solutions, and rank ranges of related quaternion matrix equations.

Contribution

It introduces a novel simultaneous decomposition technique for seven quaternion matrices and applies it to solve and analyze specific quaternion matrix equations.

Findings

Derived solvability conditions for quaternion matrix equations.

Provided general solutions for the equations.

Established rank bounds for the solutions.

Abstract

Let $\mathbb{H}$ be the real quaternion algebra and $\mathbb{H}^{n\times m}$ denote the set of all $n\times m$ matrices over $\mathbb{H}$. In this paper, we construct a simultaneous decomposition of seven general real quaternion matrices with compatible sizes: $A\in \mathbb{H}^{m\times n}, B\in \mathbb{H}^{m\times p_{1}},C\in \mathbb{H}^{m\times p_{2}},D\in \mathbb{H}^{m\times p_{3}},E\in \mathbb{H}^{q_{1}\times n},F\in \mathbb{H}^{q_{2}\times n},G\in \mathbb{H}^{q_{3}\times n}$. As applications of the simultaneous matrix decomposition, we give solvability conditions, general solutions, as well as the range of ranks of the general solutions to the following two real quaternion matrix equations $BXE+CYF+DZG=A$ and $BX+WE+CYF+DZG=A,$ where $A,B,C,D,E,F,$ and $G$ are given real quaternion matrices.