Resolvent bounds and spectral variation
type: Item
instance of: scholarly article
title: Resolvent bounds and spectral variation
zbMATH Open document ID: 0717.15016
DOI: 10.1016/0024-3795(91)90323-O
author: Dennis Phillips
published in: Linear Algebra and its Applications
publication date: 1991-01-01T00:00:00Z
review text: The spectral variation between linear operators A, B mapping an n- dimensional space over \({\mathbb{C}}\) is given by \(v(A,B)=\min_{\pi}\max_{i}| \lambda_ i-\mu_{\pi (i)}|\) where \(\pi\) varies over all permutations of \(\{\) 1,...,n\(\}\). Here \(\{\lambda_ 1,...,\lambda_ n\}\) is the spectrum of A and \(\{\mu_ 1,...,\mu_ n\}\) is the spectrum of B; \(\| \cdot \|\) is a norm when applied to a vector and the inherited operator norm when applied to a linear operator. It is shown that \(v(A,B)\leq 8\times 4^{-1/n}M^{1- 1/n}\| A-B\|^{1/n}\times [\sum^{n- 1}_{i=0}(\frac{v(A,B)}{2M})^ i]^{1/n},\) where \(M=\max \{\| A\|,\| B\| \}\). This compares to a known result \(v(A,B)\leq 8\times 4^{-1/n}n^{1/n}M^{1-1/n}\| A-B\|^{1/n}\).
Mathematics Subject Classification ID: 15A60
Mathematics Subject Classification ID: 15A45
Mathematics Subject Classification ID: 15A42
zbMATH DE Number: 4181533
zbMATH Keywords: Cayley-Hamilton theorem
zbMATH Keywords: Chebyshev polynomials
zbMATH Keywords: resolvent bounds
zbMATH Keywords: spectral variation
zbMATH Keywords: spectrum
zbMATH Keywords: operator norm
reviewed by: Richard Sinkhorn
MaRDI profile type: Publication
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full work available at URL: 0024-3795(91)90323-o
OpenAlex ID: W2165363839
description: scientific article; zbMATH DE number 4181533