A simultaneous decomposition of seven matrices over real quaternion algebra and its applications

Abstract

Let $\mathbb{H}$ be the real quaternion algebra and ${\mathbb{H}}^{n\times m}$ denote the set of all $n\times m$ matrices over $\mathbb{H}$. In this paper, we construct a simultaneous decomposition of seven general real quaternion matrices with compatible sizes: $A\in {\mathbb{H}}^{m\times n},B\in {\mathbb{H}}^{m\times p_1},C\in {\mathbb{H}}^{m\times p_2},D\in {\mathbb{H}}^{m\times p_3},E\in {\mathbb{H}}^{q_1\times n},F\in {\mathbb{H}}^{q_2\times n},G\in {\mathbb{H}}^{q_3\times n}$. As applications of the simultaneous matrix decomposition, we give solvability conditions, general solutions, as well as the range of ranks of the general solutions to the following two real quaternion matrix equations $BXE+CYF+DZG=A$ and $BX+WE+CYF+DZG=A,$ where $A,B,C,D,E,F,$ and $G$ are given real quaternion matrices.