Solving Optimization Problems on Hermitian Matrix Functions with Applications

Abstract EN

We consider the extremal inertias and ranks of the matrix expressions f(X,Y)=A3-B3X-(B3X)*-C3YD3-(C3YD3)*, where A3=A3*, B3, C3, and D3 are known matrices and Y and X are the solutions to the matrix equations A1Y=C1, YB1=D1, and A2X=C2, 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 A1Y=C1, YB1=D1, A2X=C2, and B3X+(B3X)*+C3YD3+(C3YD3)*=A3, and give an expression of the general solution to the above-mentioned system when it is solvable.