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Common Hermitian least squares solutions of matrix equations \(A_1XA^*_1=B_1\) and \(A_2XA^*_2=B_2\) subject to inequality restrictions. (English) Zbl 1231.15002
Summary: We give some closed-form formulas for calculating maximal and minimal ranks and inertias of \(P - X\) with respect to \(X\), where \(P\in C_H^n\) is given, \(X\) is a common Hermitian least squares solutions to the matrix equations \(A_1XA^*_1=B_1\) and \(A_2XA^*_2=B_2\). As an application, we derive necessary and sufficient conditions for \(X>P(\geq P,<P,\leq P)\) in the Löwner partial ordering. In addition, we give identifying conditions for the existence of definite common Hermitian least squares solutions to matrix equations \(A_1XA^*_1=B_1\) and \(A_2XA^*_2=B_2\).

15A03 Vector spaces, linear dependence, rank, lineability
15A24 Matrix equations and identities
Full Text: DOI
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