Diagonals and numerical ranges of weighted shift matrices
type: Item
instance of: scholarly article
title: Diagonals and numerical ranges of weighted shift matrices
zbMATH Open document ID: 1261.15032
author: Kuo-Zhong Wang
author: Pei Yuan Wu
published in: Linear Algebra and its Applications
publication date: 2013-01-21T00:00:00Z
review text: For an \(n\)-by-\(n\) matrix \(A\), the authors examine the maximum number \(k = k(A)\) for which there is a \(k\)-by-\(k\) compression of~\(A\) with all its diagonal entries in the boundary \(\partial W(A)\) of the numerical range \(W(A)\) of~\(A\). For any such compression, a~standard model under unitary equivalence for~\(A\) is given. This is then applied to determine the value of \(k(A)\) for \(A\) of size~\(3\) in terms of the shape of its numerical range \(W(A)\). A~case of special interest is when \(k(A)\) equals the size of~\(A\). For the special case of weighted shift matrices, a~result is obtained to exactly determine when \(k(A)\) equals~\(n\). This is so if and only if either \(| w_1|=\dotsc=| w_n|\) or \(n\) is even and \(| w_1|=| w_3|=\dotsc=| w_{n-1}|\) and \(| w_2|=| w_4|=\dotsc=| w_n|\). Also, a characterization of those matrices~\(A\) for which \(k(A)=n-1\) is put forward.
Mathematics Subject Classification ID: 15A60
Mathematics Subject Classification ID: 15A21
zbMATH DE Number: 6127356
zbMATH Keywords: numerical ranges
zbMATH Keywords: weighted shift matrix
zbMATH Keywords: compression
zbMATH Keywords: unitary equivalence
reviewed by: A. R. Alimov
MaRDI profile type: Publication
Wikidata QID: Q114152380
full work available at URL: j.laa.2012.08.007
OpenAlex ID: W1992129747
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DOI: 10.1016/J.LAA.2012.08.007
description: scientific article; zbMATH DE number 6127356