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

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.