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Дата последнего поиска статьи во внешних источниках: 27 мая 2015 г.
Аннотация:Let A be a quadratic matrix of order n with complex elements (we denote this by A∈ℂ n×n ) and UΣV * the singular value decomposition of A, where U, V are unitary matrices, Σ=diag(σ 1 (A),σ 2 (A),⋯,σ n (A)) and σ 1 (A)≥σ 2 (A)≥⋯≥σ n (A)≥0 are the singular values of A· For k=1,2,⋯,n let us consider S 1 (k) ={S∈ℂ n×n :∑σ j 1 (S)⋯σ j k (S)≤1}· The authors prove that the set S 1 (k) (1≤k≤n) is a Chebyshev set in ℂ n×n with respect to the spectral norm. One obtains the formula for the distance from A∈ℂ n×n to S 1 (k) and proves that the metric projection P S 1 (k) is globally Lipschitz-continuous.