Extremal ranks of a quaternion matrix expression subject to consistent systems of quaternion matrix equations with applications

Assume that the linear quaternion matrix expression f(X1, X2) = A − A3X1B3 − A4X2B4 where X1, X2 are variant quaternion matrices. In this paper, we derive the maximal and minimal ranks of f(X1, X2) subject to consistent systems of quaternion matrix equations A1X1 = C1, X1B1 = C2 and A2X2 = C3, X2B2 = C4. Moreover, corresponding results on some special cases are presented. As applications, we give necessary and sufficient conditions for the existence of solutions to some systems of quaternion matrix equations. Some previous known results can be regarded as the special cases of this paper.