Maximization and minimization of the rank and inertia of the Hermitian matrix expression A-BX-(BX)* with applications

We give in this paper a group of closed-form formulas for the maximal and minimal ranks and inertias of the linear Hermitian matrix function A-BX-(BX)* with respect to a variable matrix X. As applications, we derive the extremal ranks and inertias of the matrices X±X*, where X is a solution to the matrix equation AXB=C, and then give necessary and sufficient conditions for the matrix equation AXB=C to have Hermitian, definite and Re-definite solutions. In addition, we give closed-form formulas for the extremal ranks and inertias of the difference X1-X2, where X1 and X2 are Hermitian solutions of two matrix equations and , and then use the formulas to characterize relations between Hermitian solutions of the two equations.