Ziȩtak, K. Strict spectral approximation of a matrix and some related problems. (English) Zbl 0885.15016 Appl. Math. 24, No. 3, 267-280 (1997). 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. Reviewer: Alexey Alimov (Erlangen) Cited in 1 Document MSC: 15A60 Norms of matrices, numerical range, applications of functional analysis to matrix theory 15A09 Theory of matrix inversion and generalized inverses Keywords:strict spectral approximation of a matrix; \(c_ p\)-minimal approximation; singular value preserving functions; positive semi-definite matrix; Moore-Penrose generalized inverse PDFBibTeX XMLCite \textit{K. Ziȩtak}, Appl. Math. 24, No. 3, 267--280 (1997; Zbl 0885.15016) Full Text: DOI EuDML