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

Abstract EN

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.