Bisymmetric and centrosymmetric solutions to systems of real quaternion matrix equations.

*(English)*Zbl 1138.15003Summary: In this paper we consider bisymmetric and centrosymmetric solutions to certain matrix equations over the real quaternion algebra $\mathbb{H}$. Necessary and sufficient conditions are obtained for the matrix equation $AX=C$ and the following systems

$$\begin{array}{cc}\hfill {A}_{1}X& ={C}_{1},\hfill \\ \hfill X{B}_{3}& ={C}_{3},\hfill \end{array}\phantom{\rule{1.em}{0ex}}\phantom{\rule{1.em}{0ex}}\begin{array}{cc}\hfill {A}_{1}X& ={C}_{1},\hfill \\ \hfill {A}_{2}X& ={C}_{2},\hfill \end{array}$$

to have bisymmetric solutions, and the system

to have centrosymmetric solutions. The expressions of such solutions of the matrix and the systems mentioned above are also given. Moreover a criterion for a quaternion matrix to be bisymmetric is established and some auxiliary results on other sets over $\mathbb{H}$ are also mentioned.