Extreme ranks of a linear quaternion matrix expression subject to triple quaternion matrix equations with applications

Abstract

In this paper, we establish the formulas of the maximal and minimal ranks of the quaternion matrix expression C 4 - A 4 XB 4 where X is a variant quaternion matrix subject to quaternion matrix equations A 1 X = C 1 , XB 2 = C 2 , A 3 XB 3 = C 3 . As applications, we give a new necessary and sufficient condition for the existence of solutions to the system of matrix equations A 1 X = C 1 , XB 2 = C 2 , A 3 XB 3 = C 3 , A 4 XB 4 = C 4 , which was investigated by Wang [Q.W. Wang, A system of four matrix equations over von Neumann regular rings and its applications, Acta Math. Sin., 21(2) (2005) 323–334], by rank equalities. In addition, extremal ranks of the generalized Schur complement D - CA - B with respect to an inner inverse A − of A, which is a common solution to quaternion matrix equations A 1 X = C 1 , XB 2 = C 2 , are also considered. Some previous known results can be viewed as special cases of the results of this paper.