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

In this paper, we establish the formulas of the maximal and minimal ranks of the quaternion matrix expression C4-A4XB4 where X is a variant quaternion matrix subject to quaternion matrix equations A1X=C1,XB2=C2,A3XB3=C3. As applications, we give a new necessary and sufficient condition for the existence of solutions to the system of matrix equations A1X=C1,XB2=C2,A3XB3=C3,A4XB4=C4, 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 A1X=C1,XB2=C2, are also considered. Some previous known results can be viewed as special cases of the results of this paper.