A Sylvester-type matrix equation over the Hamilton quaternions with an application

TL;DR

This paper establishes solvability conditions and solutions for a Sylvester-type matrix equation over Hamilton quaternions, with applications to quaternion matrix equations involving η-Hermicity, supported by an algorithm and numerical example.

Contribution

It introduces new solvability criteria and solution formulas for quaternion Sylvester equations, extending classical results to Hamilton quaternions with practical algorithms.

Findings

Derived solvability conditions for quaternion Sylvester equations

Provided explicit solution formulas for these equations

Developed an algorithm with numerical illustration

Abstract

We derive the solvability conditions and a formula of a general solution to a Sylvester-type matrix equation over Hamilton quaternions. As an application, we investigate the necessary and sufficient conditions for the solvability of the quaternion matrix equation, which involves $\eta$-Hermicity. We also provide an algorithm with a numerical example to illustrate the main results of this paper.