The numerical range of derivations
type: Item
instance of: scholarly article
title: The numerical range of derivations
zbMATH Open document ID: 0676.15013
DOI: 10.1016/0024-3795(89)90071-2
author: Chi-Kwong Li
author: Nam-Kiu Tsing
published in: Linear Algebra and its Applications
publication date: 1989-01-01T00:00:00Z
review text: Let p, q, n be integers satisfying \(1\leq p\leq q\leq n\). The (p,q)- numerical range of an \(n\times n\) complex matrix A is defined by \(W_{p,q}(A)=\{E_ p((UAU^*)[q]):\) U unitary\(\}\), where for an \(n\times n\) complex matrix X, X[q] denotes its \(q\times q\) leading principal submatrix and \(E_ p(X[q])\) denotes the pth elementary symmetric function of the eigenvalues of X[q]. The authors prove that if \(2\leq p\leq q\leq n-2\), then \(W_{p,q}(A)\) is not convex for the normal matrix \(A=I_{n-2}\oplus (1+i)I_ 2.\) Combining with known results, this shows that \(W_{p,q}(A)\) is convex for all A if and only if \(p=1\) or \(q\geq n-1\). Then they study various inclusion relations and inequalities involving \(r_{p,q}(A)=\max \{| \mu |:\quad \mu \in W_{p,q}(A)\}.\) More detailed description is obtained when A belongs to various classes of matrices.
Mathematics Subject Classification ID: 15A60
zbMATH DE Number: 4107016
zbMATH Keywords: numerical range
zbMATH Keywords: numerical radius
zbMATH Keywords: eigenvalues
zbMATH Keywords: convex
zbMATH Keywords: normal matrix
reviewed by: Yiu-Tung Poon
MaRDI profile type: Publication
full work available at URL: 0024-3795(89)90071-2
OpenAlex ID: W2003839345
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description: scientific article; zbMATH DE number 4107016