A simultaneous decomposition of five real quaternion matrices with applications

TL;DR

This paper develops a simultaneous decomposition method for five real quaternion matrices, enabling analysis of their ranks, solutions to matrix equations, and extending to seven matrices, with applications in solving quaternion matrix equations.

Contribution

It introduces a novel simultaneous decomposition technique for multiple quaternion matrices, facilitating rank analysis and solutions to matrix equations.

Findings

Derived maximal and minimal ranks of quaternion matrix expressions

Established solvability conditions for quaternion matrix equations

Provided general solutions to specific quaternion matrix equations

Abstract

In this paper, we construct a simultaneous decomposition of five real quaternion matrices in which three of them have the same column numbers, meanwhile three of them have the same row numbers. Using the simultaneous matrix decomposition, we derive the maximal and minimal ranks of some real quaternion matrices expressions. We also show how to choose the variable real quaternion matrices such that the real quaternion matrix expressions achieve their maximal and minimal ranks. As an application, we give a solvability condition and the general solution to the real quaternion matrix equation $BXD+CYE=A$. Moreover, we give a simultaneous decomposition of seven real quaternion matrices.