Solving optimization problems on hermitian matrix functions with applications

We consider the extremal inertias and ranks of the matrix expressions f (x,y) = a 3 - b 3 x - (b 3 x) - c 3 y d 3 - (c 3 y d 3) where a 3 = a 3 b 3,c 3,and d 3 are known matrices and y and x are the solutions to the matrix equations a 1 y = c 1,y b 1 = d 1,and 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,y b 1 = d 1,a 2 x = c 2,and b 3 x + (b 3 x) + c 3 y d 3 + (c 3 y d 3) = a 3,and give an expression of the general solution to the above-mentioned system when it is solvable. © 2013 xiang zhang and shu-wen xiang.