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. © 2013 yirong yao.