Copyright © 2013 Xiang Zhang and Shu-Wen Xiang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We consider the extremal inertias and ranks of the matrix expressions , where , and are known matrices and and are the solutions to the matrix equations , , and , respectively. As applications, we present necessary and sufficient condition for the previous matrix function 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 , , , and , and give an expression of the general solution to the above-mentioned system when it is solvable.