Equality of higher numerical ranges of matrices and a conjecture of Kippenhahn on Hermitian pencils
type: Item
instance of: scholarly article
title: Equality of higher numerical ranges of matrices and a conjecture of Kippenhahn on Hermitian pencils
zbMATH Open document ID: 0929.15024
DOI: 10.1016/S0024-3795(97)00251-6
author: Chi-Kwong Li
author: Ilya Spitkowsky
author: Sudheer Shukla
published in: Linear Algebra and its Applications
publication date: 1999-02-23T00:00:00Z
review text: For an \(n\)-by-\(n\) complex matrix \(A\) and \(1\leq k\leq n\) the \(k\)th numerical range of \(A\) is defined by \(W_k(A)=\{(1/k) \sum^k_{j=1}x^*_jAx_j: \{x_1,\dots,x_n\}\) is an orthonormal set in \({\mathbb C}^n\}\). When \(k=1\), \(W_k(A)\) reduces to the classical numerical range. It is known that \(\{\text{tr}A/n\}= W_n(A)\subseteq W_{n-1}(A)\subseteq\dots\subseteq W_1(A)\). The purpose of this paper is to study conditions on \(A\) so that \(W_m(A)=W_k(A)\) for some given \(m,k\). It turns out that this study is related to a conjecture of \textit{R. Kippenhahn} [Math. Nachr. 6, 193-228 (1951; Zbl 0044.16201)]. As a by-product the authors obtain a new class of \(6\times 6\) counterexamples to the Kippenhahn conjecture.
Mathematics Subject Classification ID: 15A60
Mathematics Subject Classification ID: 15A42
Mathematics Subject Classification ID: 15A22
zbMATH DE Number: 1109507
zbMATH Keywords: simultaneous contractibility
zbMATH Keywords: joint spectral radius
zbMATH Keywords: optimal joint spectral radius range
zbMATH Keywords: positive definite matrices
zbMATH Keywords: Rota's theorem
zbMATH Keywords: John's ellipsoid method
zbMATH Keywords: Hermitian pencils
zbMATH Keywords: numerical range
zbMATH Keywords: counterexamples
reviewed by: A. R. Alimov
MaRDI profile type: Publication
Wikidata QID: Q123148580
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description: scientific article; zbMATH DE number 1109507