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

A simultaneous decomposition of five real quaternion matrices with applications