Numerical ranges and Geršgorin discs
type: Item
instance of: scholarly article
title: Numerical ranges and Geršgorin discs
zbMATH Open document ID: 1259.15034
author: Chi-Tung Chang
author: Hwa-Long Gau
author: Kuo-Zhong Wang
author: Pei Yuan Wu
published in: Linear Algebra and its Applications
publication date: 2013-01-16T00:00:00Z
review text: Let \(A=[a_{ij}]\) be an \(n\times n\) complex matrix and \(W(A)\) be its numerical range. Define the Geršgorin region \(G(A)\) to be the convex hull of \(\cup_{i=1}^n\{z \in \mathbb{C}: |z-a_{ii}| \leq \left(\sum_{i\neq j}(|a_{ij}|+|a_{ji}|)\right)/2\}\) and the unitarily reduced Geršgorin region \(G'(A)\) to be \(\cap\{G(U^*AU): U \text{ is an } n\times n \text{ unitary matrix}\}\). It is known that \(W(A)\) is contained in \(G(A)\) and hence in \(G'(A)\), see \textit{C. Johnson} [Proc. Am. Math. Soc. 41, 57--60 (1973; Zbl 0248.15014)]. In the paper under review the authors present a decomposition theorem and use it to discuss conditions for \(W(A)\) to be equal to \(G(A)\) or \(G'(A)\). They show that if \(W(A) = G'(A)\), then the boundary of \(W(A)\) consists only of circular arcs and line segments. If, moreover, \(A\) is unitarily irreducible, then \(W(A)\) is a circular disc. They also give criteria for the equality of \(W(A)\) and \(G(A)\). In particular, it is shown that such \(A\)'s among the permutationally irreducible ones must have even sizes. They also characterize those \(A\)'s with size \(2\) or \(4\) satisfying \(W(A) = G(A)\) by showing that this occurs if and only if \(A\) is permutationally similar to a certain direct sum.
reviewed by: Mohammad Sal Moslehian
Mathematics Subject Classification ID: 15A60
Mathematics Subject Classification ID: 15A42
zbMATH DE Number: 6125856
zbMATH Keywords: numerical range
zbMATH Keywords: Geršgorin discs
zbMATH Keywords: unitarily irreducible matrix
zbMATH Keywords: permutationally irreducible matrices
MaRDI profile type: Publication
full work available at URL: j.laa.2012.09.003
OpenAlex ID: W2074868931
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DOI: 10.1016/J.LAA.2012.09.003
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description: scientific article; zbMATH DE number 6125856