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The minimum norm solutions of two classes of matrix equations. (Chinese. English summary) Zbl 1021.15010

Summary: The following problems are considered:
Problem I: Given \(A\in\mathbb{R}^{m\times n}\), \(B\in\mathbb{R}^{p\times n}\), \(C\in \mathbb{R}^{n\times n} \). Let \(S_1=\{X:X\in \mathbb{R}^{m\times p}\), \(A^TXB+ B^TX^TA=C\}\). Find \(\widehat X \in S_1\) such that \(\|\widehat X\|=\min\).
Problem II: Given \(A\in \mathbb{R}^{m\times n}\), \(B\in\mathbb{R}^{m\times p}\), \(C\in\mathbb{R}^{n\times n}\), \(D\in \mathbb{R}^{p\times p}\). Let \(S_2=\{X: X\in SR^{m\times m}\), \(A^TXA=C\), \(B^TXB=D\}\). Find \(\widehat X\in S_2\) such that \(\|\widehat X\|= \min\).
By applying the generalized singular-value and canonical correlation decompositions of matrix pairs, necessary and sufficient conditions under which \(S_1\) and \(S_2\) are nonempty are studied. Expressions for the solutions of problems I and II are given.

MSC:

15A24 Matrix equations and identities
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