The general solutions to some systems of Sylvester-type quaternion matrix equations with an application

TL;DR

This paper establishes conditions for the solvability of Sylvester-type quaternion matrix equations and derives their general solutions, with applications to quaternion matrix equations involving $\

Contribution

It provides a comprehensive framework for solving Sylvester-type quaternion matrix equations, including necessary and sufficient conditions and explicit general solutions.

Findings

Derived necessary and sufficient conditions for system consistency

Presented explicit formulas for general solutions

Provided an algorithm and example illustrating the results

Abstract

Sylvester-type matrix equations have applications in areas including control theory, neural networks, and image processing. In this paper, we establish the necessary and sufficient conditions for the system of Sylvester-type quaternion matrix equations to be consistent and derive an expression of its general solution (when it is solvable). As an application, we investigate the necessary and sufficient conditions for quaternion matrix equations to be consistent and derive a formula for its general solution involving $\eta$-Hermicity. As a special case, we also present the necessary and sufficient conditions for the system of two-sided Sylvester-type quaternion matrix equations to have a solution and derive a formula for its general solution (when it is solvable). Finally, we present an algorithm and an example to illustrate the main results of this paper.