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Strict spectral approximation of a matrix and some related problems. (English) Zbl 0885.15016

Let \(M\) be a non-empty closed convex subset of the normed linear space \(\mathbb{C}^{m\times n}\) of \(m\times n\) complex matrices. Given a matrix \(A\in\mathbb{C}^{m\times n}\) the problem \(\min_{X\in M}|A-X|_\infty\) is considered, where \(|\cdot|_\infty\) is the spectral norm: \(|A|_\infty=\sigma_1(A)\), the maximal singular value of \(A\).
A matrix \(\widehat A\) is a strict spectral approximation of \(A\) from \(M\), if the vector \(\sigma(A-\widehat A)\) of decreasing ordered singular values of \((A-\widehat A)\) is minimal with respect to the lexicographic ordering \(\leqslant_\ell\) in the set \(\{\sigma\in\mathbb{R}^q_+\downarrow:\sigma=\sigma(A-X)\), \(X\in M\), \(q=\min(m,n)\}\).
Theorem. A matrix \(\widehat X\) is a strict spectral approximant to \(A\) from \(M\) if and only if for every \(X\in M\), \(X\neq\widehat X\), we have \(|A-X|_p>|A-\widehat X|_p\) for all \(p>1\) sufficiently large.
The author studies properties of approximations from linear subspaces described by linear, singular values preserving functions and gives necessary conditions and sufficient conditions for uniqueness. Some characterisations of Moore-Penrose generalised inverse matrices are also obtained.

MSC:

15A60 Norms of matrices, numerical range, applications of functional analysis to matrix theory
15A09 Theory of matrix inversion and generalized inverses
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