Condition numbers of matrices with given spectrum
type: Item
instance of: scholarly article
title: Condition numbers of matrices with given spectrum
zbMATH Open document ID: 1430.15014
author: Rachid Zarouf
author: Stéphane Charpentier
author: Karine Fouchet
author: Oleg Szehr
published in: Analysis and Mathematical Physics
publication date: 2019-11-28T00:00:00Z
review text: This paper surveys recent strategies to estimate the condition number \(\operatorname{CN}(T)=||T||||T^{-1}||\) of complex \(n\times n\) matrices \(T\) with a given spectrum. More precisely, the authors present a proof of the fact that if \(T\) acts on the Hilbert space \(C_n\), then the supremum of \(\operatorname{CN}(T)\) over all contractions \(T\) with smallest eigenvalues of modulus \(r > 0\), is equal to \(1/r_n\). This is achieved using an analytic Toeplitz matrix. The same question is posed for \(n\)-dimensional Banach spaces. These strategies provide explicit and constructive solutions to the so-called Halmos and Schäffer problems and are also shown to be effective in a closely related situation, namely considering Kreiss matrices instead of contractions.
reviewed by: V. Lokesha
Mathematics Subject Classification ID: 15A60
Mathematics Subject Classification ID: 15B05
Mathematics Subject Classification ID: 47A10
zbMATH DE Number: 7138524
zbMATH Keywords: Toeplitz matrices
zbMATH Keywords: condition numbers
zbMATH Keywords: model operator
zbMATH Keywords: Blaschke product
zbMATH Keywords: Halmos/Schäffer problems
MaRDI profile type: Publication
full work available at URL: s13324-019-00328-4
OpenAlex ID: W2945837529
cites work: On some power sum problems of Turán and Erdös
cites work: A probabilistic approach to problems of diophantine approximation
cites work: Zeros of analytic functions and norms of inverse matrices
cites work: On the Eigenvalues of a Matrix with Prescribed Singular Values
cites work: Analytical inversion of symmetric tridiagonal matrices
cites work: Q4315110
cites work: Condition numbers of large matrices, and analytic capacities
cites work: Q4537721
cites work: Q3945080
cites work: Q4486034
cites work: Q5688384
cites work: Q3142970
cites work: Norms and determinants of linear mappings
cites work: Eigenvalue estimates for the resolvent of a non-normal matrix
cites work: Maximum of the resolvent over matrices with given spectrum
cites work: Explicit counterexamples to Schäffer's conjecture
cites work: Q5600360
cites work: Q3333131
cites work: Functional calculus under Kreiss type conditions
cites work: Toeplitz condition numbers as an \(H^{\infty }\) interpolation problem
DOI: 10.1007/S13324-019-00328-4
description: scientific article; zbMATH DE number 7138524