On eigenvalues and boundary curvature of the numerical range
type: Item
instance of: scholarly article
title: On eigenvalues and boundary curvature of the numerical range
zbMATH Open document ID: 0969.15014
DOI: 10.1016/S0024-3795(00)00231-7
author: Lauren Caston
author: Milena Savova
author: Nahum Zobin
author: Ilya M. Spitkovsky
published in: Linear Algebra and its Applications
publication date: 2001-09-27T00:00:00Z
review text: For an \(n\times n\) matrix \(A\) let \(W(A)\) be its numerical range, and \(\sigma (A)\) be its spectrum. By the theorem of \textit{W. F. Donoghue jun.} [Michigan Math. J. 4, 261-263 (1958; Zbl 0082.11601)], all corner points of \(W(A)\) belong to~\(\sigma (A)\). The authors obtain bounds for the distance \(D_p(A)\) from a point~\(p\) on the boundary \(\partial W(A)\) to \( \sigma (A)\). Let \(M(A)\) be the smallest constant such that \(D_p(A)\leqslant M(A)R_p(A)\) for all \(p\in \partial W(A)\), where \(R_p(A)\) is the radius of curvature of~\(\partial W(A)\), being defined for all but finitely many points, and let \(M_n=\sup \{M(A) : A\in {\mathbb C}^{n\times n}\}\). The main result is the following inequality: \(\frac n2\sin \frac \pi n\leqslant M_n\leqslant \frac n2\), which becomes an identity for \(n=2\). A description of some matrices \(A\) where \(M(A)=1\), as well as some computations of exact values of \(M(A)\) for \(2\times 2\) matrices \(A\) is given.
Mathematics Subject Classification ID: 15A60
Mathematics Subject Classification ID: 15A42
zbMATH DE Number: 1559345
zbMATH Keywords: numerical range
zbMATH Keywords: boundary curvature
zbMATH Keywords: inequalities involving eigenvalues
zbMATH Keywords: spectrum
reviewed by: A. R. Alimov
MaRDI profile type: Publication
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Wikidata QID: Q126666602
description: scientific article; zbMATH DE number 1559345