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On the Hermitian structures of the solution to a pair of matrix equations. (English) Zbl 1264.15020

Linear Multilinear Algebra 61, No. 1, 73-90 (2013); corrigendum ibid. 61, No. 8, 1158 (2013).
Let \(\mathcal{S}\) be a given set consisting of some Hermitian matrices with the same sizes. A matrix \(A \in \mathcal{S}\) is said to be minimal (maximal) if \(A - W\) is negative (positive) semidefinite for every matrix \(W \in \mathcal{S}\). In this article, the new quasi-quadratic Hermitian structure for a system of matrix equations is constructed, i.e., \[ \begin{aligned} p(X) &= X X^{\ast} - P, \tag{1}\\ q(X) &= X^{\ast} X - Q, \tag{2}\end{aligned} \] where \(P = P^{\ast}, \; Q = Q^{\ast}\) and \(X\) is a solution to \[ A\,X = C, \; X\,B = D.\tag{3} \] First, the extremal inertias and ranks of (1) and (2) are considered. As applications, the necessary and sufficient conditions for (1) and (2) are derived to be positive (negative), positive (negative) semidefinite, nonsingular and the system \[ A \, X = C, \; X \, B = D, \; X \, X^{\ast} = P, \] to be consistent, respectively. Some special cases such as the unitary solvability and the contraction solvability to (3) are also considered. In addition, the necessary and sufficient conditions for the existence of the left and right minimal solutions to (3) are studied. The explicit expressions of the left and the right minimal solutions are given when the conditions are met.

MSC:

15A24 Matrix equations and identities
15A03 Vector spaces, linear dependence, rank, lineability
15A09 Theory of matrix inversion and generalized inverses
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