Numerical ranges of companion matrices
type: Item
instance of: scholarly article
title: Numerical ranges of companion matrices
zbMATH Open document ID: 1128.15017
author: Hwa-Long Gau
author: Pei Yuan Wu
published in: Linear Algebra and its Applications
publication date: 2007-03-09T00:00:00Z
review text: The authors show that an \(n\)-by-\(n\) companion matrix \(A\) can have at most \(n\) line segments on the boundary \(\partial W(A)\) of its numerical range \(W(A)\), and it has exactly \(n\) line segments on \(\partial W(A)\) if and only if, for \(n\) odd, \(A\) is unitary, and, for \(n\) even, \(A\) is unitarily equivalent to the direct sum \(A_1\oplus A_2\) of two \((n/2)\)-by-\((n/2)\) companion matrices \(A_1\) and \(A_2\) with \(1\leqslant | A| <\tan (\pi/n) + \sec (\pi/n)\). Here for both \(A_1\) and \(A_2\) the main diagonal is totally 0, their superdiagonal is totally 1, and the lower-left entry of \(A_1\) is \(a\), and \(-1/\bar a\), for \(A_2\).
Mathematics Subject Classification ID: 15A60
zbMATH DE Number: 5132591
zbMATH Keywords: numerical range
zbMATH Keywords: boundary
zbMATH Keywords: unitary matrix
zbMATH Keywords: companion matrix
reviewed by: A. R. Alimov
MaRDI profile type: Publication
full work available at URL: j.laa.2006.03.037
OpenAlex ID: W1974195407
cites work: On flat portions on the boundary of the numerical range
cites work: The numerical range of an operator
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cites work: Companion matrices: Reducibility, numerical ranges and similarity to contractions
cites work: The Numerical Radius of a Nilpotent Operator on a Hilbert Space
cites work: Q4040931
cites work: Q4105628
cites work: On matrices whose numerical ranges have circular or weak circular symmetry
DOI: 10.1016/J.LAA.2006.03.037
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description: scientific article; zbMATH DE number 5132591