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

Abstract

We solve optimization problems on the ranks and inertias of the quadratic Hermitian matrix function $Q-XP{X}^{*}$ 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 in the Löwner partial ordering to be feasible, respectively. The findings of this paper widely extend the known results in the literature.