Solutions to optimization problems on ranks and inertias of a matrix function with applications

Consider the Hermitian matrix function f(X)=A3-B3X-(B3X)∗ subject to a consistent system of matrix equations (0.1)A1X=C1,A2XB2=C2,where ∗ means conjugate transpose. In this paper we first establish explicit expansion formulas to calculate the global maximal and minimal ranks and inertias of the Hermitian matrix function f(X), then we use the derived formulas to give necessary and sufficient conditions for system (0.1) to have Re-nonnegative definite, Re-nonpositive definite, Re-positive definite, and Re-negative definite solutions. Moreover, as another application of the derived formulas, we establish necessary and sufficient conditions for the solvability to the system of matrix equations (0.2)A1X=C1,A2XB2=C2,B3X+(B3X)∗=A3and provide an expression of the general solution to (0.2) when it is solvable. The findings of this paper widely extend the known results in the literature.