A constrained system of matrix equations

Sylvester-type matrix equations have wide applications in system science, control theory, and so on. In this paper, we consider a constrained system of Sylvester-type matrix equations over the quaternions. We first derive the necessary and sufficient conditions for the system to have a solution and propose a formula of its general solution when it is solvable. As an application, we then investigate the η\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\eta $$\end{document}-Hermitian solution of a system. Moreover, we also give a numerical example to illustrate the main findings of this paper.