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We solve optimization problems on the ranks and inertias of the quadratic Hermitian matrix function  subject to a consistent system of matrix equations  and . As applications, we derive necessary and sufficient conditions for the solvability to the systems of matrix equations and matrix inequalities , and  in the Löwner partial ordering to be feasible, respectively. The findings of this paper widely extend the known results in the literature

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