Solving Optimization Problems on Hermitian Matrix Functions with Applications

Abstract

We consider the extremal inertias and ranks of the matrix expressions $f(X,Y)=A_{\mathrm{3}}-B_{\mathrm{3}}X-(B_{\mathrm{3}}X{)}^{\mathrm{*}}-C_{\mathrm{3}}Y{D}_{\mathrm{3}}-(C_{\mathrm{3}}Y{D}_{\mathrm{3}}{)}^{\mathrm{*}}$, where $A_{\mathrm{3}}=A_{\mathrm{3}}^{\mathrm{*}},$ $B_{\mathrm{3}},$ $C_{\mathrm{3}}$, and $D_{\mathrm{3}}$ are known matrices and $Y$ and $X$ are the solutions to the matrix equations $A_{\mathrm{1}}Y=C_{\mathrm{1}}$, $Y{B}_{\mathrm{1}}=D_{\mathrm{1}}$, and $A_{\mathrm{2}}X=C_{\mathrm{2}}$, respectively. As applications, we present necessary and sufficient condition for the previous matrix function $f(X,Y)$ to be positive (negative), non-negative (positive) definite or nonsingular. We also characterize the relations between the Hermitian part of the solutions of the above-mentioned matrix equations. Furthermore, we establish necessary and sufficient conditions for the solvability of the system of matrix equations $A_{\mathrm{1}}Y=C_{\mathrm{1}}$, $Y{B}_{\mathrm{1}}=D_{\mathrm{1}}$, $A_{\mathrm{2}}X=C_{\mathrm{2}}$, and $B_{\mathrm{3}}X+(B_{\mathrm{3}}X{)}^{\mathrm{*}}+C_{\mathrm{3}}Y{D}_{\mathrm{3}}+(C_{\mathrm{3}}Y{D}_{\mathrm{3}}{)}^{\mathrm{*}}=A_{\mathrm{3}}$, and give an expression of the general solution to the above-mentioned system when it is solvable.