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

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 $η$-Hermicity. We also provide an algorithm with a numerical example to illustrate the main results of this paper.

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