Ranks of a constrained hermitian matrix expression with applications

We establish the formulas of the maximal and minimal ranks of the quaternion hermitian matrix expression c4 - a4xa 4 * where x is a hermitian solution to quaternion matrix equations a1x = c1,xb1 = c2,and a3xa3 * = c3. as applications,we give a new necessary and sufficient condition for the existence of hermitian solution to the system of matrix equations a1x = c 1,xb1 = c2,a3xa3 * = c3,and a4xa4 * = c4,which was investigated by wang and wu,2010,by rank equalities. in addition,extremal ranks of the generalized hermitian schur complement c4 - a4 a3∼ a4 * with respect to a hermitian g-inverse a3∼ of a 3,which is a common solution to quaternion matrix equations a 1x = c1 and xb1 = c2,are also considered. © 2013 shao-wen yu.