Yamamoto's theorem for generalized singular values
type: Item
instance of: scholarly article
title: Yamamoto's theorem for generalized singular values
zbMATH Open document ID: 0703.15010
DOI: 10.1016/0024-3795(90)90289-O
author: Charles R. Johnson
author: Peter M. Nylen
published in: Linear Algebra and its Applications
publication date: 1990-01-01T00:00:00Z
review text: Replacing the norm by some other geometric quantities in the spectral radius formula one can obtain the moduli of all eigenvalues of a given matrix. With the singular values this was done by \textit{T. Yamamoto} [J. Math. Soc. Japan 19, 173-178 (1967; Zbl 0152.014)], and with abstract s- numbers by \textit{H. König} [Studia Math. 65, 141-146 (1979; Zbl 0432.47010)], even for compact operators on Banach spaces. These results, however, are not deeper than the spectral radius formula; they are consequences of appropriate inequalities between the geometric quantities in question and the norms of suitable restrictions. In the paper under review, the authors prove such inequalities in the finite-dimensional case by exploiting the triangular form of the operator. The main results are, of course, particular cases of H. König's theorem (loc. cit.). A more general context is shown by the reviewer [Studia Math. 80, 219-234 (1984; Zbl 0542.47008)].
Mathematics Subject Classification ID: 15A18
Mathematics Subject Classification ID: 47A10
zbMATH DE Number: 4152592
zbMATH Keywords: asymptotic formulae
zbMATH Keywords: norm
zbMATH Keywords: spectral radius formula
zbMATH Keywords: moduli of all eigenvalues
zbMATH Keywords: singular values
zbMATH Keywords: compact operators on Banach spaces
reviewed by: Jaroslav Zemánek
MaRDI profile type: Publication
cites work: Matrix Analysis
cites work: Q5343739
cites work: Q5558182
cites work: s-Numbers of operators in Banach spaces
cites work: Interlacing inequalities for generalized singular values
cites work: Powers of a matrix and a formula for the moduli of its eigenvalues
cites work: On the extreme values of the roots of matrices
full work available at URL: 0024-3795(90)90289-o
OpenAlex ID: W2059148225
description: scientific article; zbMATH DE number 4152592