A simultaneous decomposition of five real quaternion matrices with applications

In this paper, we construct a simultaneous decomposition of five real quaternion matrices in which three of them have the same column numbers, meanwhile three of them have the same row numbers. Using the simultaneous matrix decomposition, we derive the maximal and minimal ranks of some real quaternion matrices expressions. We also show how to choose the variable real quaternion matrices such that the real quaternion matrix expressions achieve their maximal and minimal ranks. As an application, we give a solvability condition and the general solution to the real quaternion matrix equation $BXD+CYE=A$. Moreover, we give a simultaneous decomposition of seven real quaternion matrices.