### The $\eta$-Hermitian solutions to some systems of real quaternion matrix equations

#### Abstract

Let $\mathbb{H}^{m\times n}$ be the set of all

$m\times n$ matrices over the real quaternion algebra. We call that $A\in\mathbb{H}^{n\times

n}$ is $\eta$-Hermitian if $A=A^{\eta*}$, where $A^{\eta*}=-\eta A^{\ast}\eta$, $\eta\in\{\mathbf{i},\mathbf{j},\mathbf{k}\}$, $\mathbf{i},\mathbf{j},\mathbf{k}$ are the quaternion units. In this paper, we derive some solvability conditions

and the general solution to a system of real quaternion matrix

equations. As an application, we present some necessary and sufficient

conditions for the existence of an $\eta$-Hermitian solution to some systems of real quaternion matrix equations. We also give the expressions of the general

$\eta$-Hermitian solutions to these systems when they are solvable. Some numerical examples are given to illustrate the results of this paper.

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