Some extremal problems for strictly totally positive matrices
type: Item
instance of: scholarly article
title: Some extremal problems for strictly totally positive matrices
zbMATH Open document ID: 0553.15008
DOI: 10.1016/0024-3795(85)90272-1
published in: Linear Algebra and its Applications
publication date: 1985-01-01T00:00:00Z
review text: For Hilbert space operators it is well known that the sequences of approximation numbers, Gelfand numbers and Bernstein numbers coincide. In the case of finite dimensional real \(\ell_ p\)-spaces (1\(\leq p\leq \infty)\) the author proves the same property for strictly totally positive matrices. He also shows that the sequences in question are strictly decreasing and identifies the extremal subspaces. The results are obtained more generally for rectangular matrices. The proof depends on studying partial derivatives of the function \(F(x)=\| Ax\|_ p/\| x\|_ p\) and sign changes in the vector x.
Mathematics Subject Classification ID: 15B48
Mathematics Subject Classification ID: 15A18
zbMATH DE Number: 3880878
zbMATH Keywords: singular numbers
zbMATH Keywords: totally positive matrices
zbMATH Keywords: approximation numbers
zbMATH Keywords: Gelfand numbers
zbMATH Keywords: Bernstein numbers
author: Allan Pinkus
reviewed by: Jaroslav Zemánek
MaRDI profile type: Publication
full work available at URL: 0024-3795(85)90272-1
OpenAlex ID: W2108633896
cites work: Q3261425
cites work: Q5624248
cites work: Spline spaces are optimal for \(L^2\) n-width
cites work: On n-Widths in L ∞
cites work: Matrices and n-widths
description: scientific article; zbMATH DE number 3880878