Solving Optimization Problems on Hermitian Matrix Functions with Applications

A 2 X = C 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 1 Y = C 1 , YB 1 = D 1 , A 2 X = C 2 , and B 3 X + (B 3 X)∗ + C 3 YD 3 + (C 3 YD 3 )∗ = A 3 , and give an expression of the general solution to the above-mentioned system when it is solvable.