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

Abstract

Assume that the linear quaternion matrix expression f( X 1, X 2) = A − A 3 X 1 B 3 − A 4 X 2 B 4 where X 1, X 2 are variant quaternion matrices. In this paper, we derive the maximal and minimal ranks of f( X 1, X 2) subject to consistent systems of quaternion matrix equations A 1 X 1 = C 1, X 1 B 1 = C 2 and A 2 X 2 = C 3, X 2 B 2 = C 4. 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.