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A Hermitian least squares solution of the matrix equation \(AXB=C\) subject to inequality restrictions. (English) Zbl 1268.15012

Summary: We give some closed-form formulas for computing the maximal and minimal ranks and inertias of \(P-X\) with respect to \(X\), where \(P\in \mathbb{C}_H^n\) is given, and \(X\) is a Hermitian least squares solution to the matrix equation \(AXB=C\). We derive, as applications, necessary and sufficient conditions for \(X\geq\) \((\leq,>,<)P\), in the Löwner partial ordering. In addition, we give necessary and sufficient conditions for the existence of a Hermitian positive (negative, nonpositive, nonnegative) definite least squares solution to \(AXB=C\).

MSC:

15A24 Matrix equations and identities
15B57 Hermitian, skew-Hermitian, and related matrices
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