On the Hermitian structures of the solution to a pair of matrix equations
Qing‐Wen Wang, Xiang Zhang, Zhuo‐Heng He · Linear and Multilinear Algebra · 2012
Let 𝒮 be a given set consisting of some Hermitian matrices with the same sizes. We say that a matrix A ∈ S is minimal (maximal) if A − W is negative (positive) semidefinite for every matrix W ∈ 𝒮. In this article, we construct the new quasi-quadratic Hermitian structures for a system of matrix equations. i.e., where P = P*, Q = Q* and X is a solution to We first consider the extremal inertias and ranks of (1) and (2). As applications, we derive the necessary and sufficient conditions for (1) and (2) to be positive (negative), positive (negative) semidefinite, nonsingular and the system to be consistent, respectively. Some special cases such as the unitary solvability and the contraction solvability to (3) are also considered. In addition, we present the necessary and sufficient conditions for the existence of the left and the right minimal solutions to (3). The explicit expressions of the left and the right minimal solutions are given when the conditions are met.