The Optimization on Ranks and Inertias of a Quadratic Hermitian Matrix Function and Its Applications

We solve optimization problems on the ranks and inertias of the quadratic Hermitian matrix function Q − XPX∗ subject to a consistent system of matrix equations AX = C and XB = D. As applications, we derive necessary and sufficient conditions for the solvability to the systems of matrix equations and matrix inequalities AX = C,XB = D, and XPX∗ = (>, <, ≥, ≤)Q in the Löwner partial ordering to be feasible, respectively. The findings of this paper widely extend the known results in the literature.