A simultaneous decomposition of four real quaternion matrices encompassing $\eta$-Hermicity and its applications

Let $\mathbb{H}$ be the real quaternion algebra and $\mathbb{H}^{m\times n}$ denote the set of all $m\times n$ matrices over $\mathbb{H}$. Let $\mathbf{i},\mathbf{j},\mathbf{k}$ be the imaginary quaternion units. For $\eta\in\{\mathbf{i},\mathbf{j},\mathbf{k}\}$, a square real quaternion matrix $A$ is said to be $\eta$-Hermitian if $A^{\eta*}=A$ where $A^{\eta*}=-\eta A^{\ast}\eta$, and $A^{\ast}$ stands for the conjugate transpose of $A$. In this paper, we construct a simultaneous decomposition of four real quaternion matrices with the same row number $(A,B,C,D),$ where $A=A^{\eta*}\in \mathbb{H}^{m\times m}, B\in \mathbb{H}^{m\times p_{1}},C\in \mathbb{H}^{m\times p_{2}},D\in \mathbb{H}^{m\times p_{3}}$. As applications of this simultaneous matrix decomposition, we derive necessary and sufficient conditions for some real quaternion matrix equations involving $\eta$-Hermicity in terms of ranks of the coefficient matrices. We also present the general solutions to these real quaternion matrix equations. Moreover, we provide some numerical examples to illustrate our results.