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The expansion problem of anti-symmetric matrix under a linear constraint and the optimal approximation. (English) Zbl 1103.15008

Denote by \(\mathbb R^{n\times m}\), \(AS\mathbb R^{n\times n}\) and \(\| .\| \) respectively the set of \(n\times m\) real matrices, the set of \(n\times n\) anti-symmetric (i.e. real skew symmetric) matrices and the Frobenius norm. The authors give necessary and sufficient conditions for the solvability and the expression of the general solution of
Problem I. Given \(A\in \mathbb R^{n\times m}\), \(B\in \mathbb R^{m\times m}\) and \(X_0\in AS\mathbb R^{q\times q}\), find \(X\in AS\mathbb R^{n\times n}\) such that \(A^TXA=B\) and the \(q\times q\) leading principal submatrix of \(X\) equals \(X_0\).
The authors solve and provide a numerical example illustrating the solution of
Problem II: Given \(X^*\in \mathbb R^{n\times n}\), find \(\hat{X}\) from the solution set \(S_E\) of Problem I such that \(\| X^*-\hat{X}\| =\min _{X\in S_E}\| X^*-X\| \).

MSC:

15A24 Matrix equations and identities
15B57 Hermitian, skew-Hermitian, and related matrices
65F30 Other matrix algorithms (MSC2010)
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References:

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