The numerical range of products of normal matrices
type: Item
instance of: scholarly article
title: The numerical range of products of normal matrices
zbMATH Open document ID: 0937.15022
DOI: 10.1016/S0024-3795(98)10224-0
published in: Linear Algebra and its Applications
publication date: 2000-02-28T00:00:00Z
review text: Let \(f_i,g_i\), \(1\leq i\leq n\) be complex numbers, \(\Sigma_n\) the simplex of nonnegative \(n\)-tuples summing to unity. Define for \(t\in \Sigma_n\) the functions \(f(t)=\sum_j t_i f_i\), \(\varphi (t)=\sqrt{\sum_{i<j} t_i t_j |f_i-f_j|^2}.\) Define \(g(s)\) and \(\psi(s)\) similarly for \(s\in \Sigma_n\) and the \(g_i.\) For \(X,Y\subseteq \{1,\ldots,n\}\) let \({\mathcal C}(X,Y):=\{z\in \mathbb C: |z-f(t)g(s)|\leq \varphi(t) \psi(s)\); \(\text{supp}(t)\subseteq X\), \(\text{supp}(s)\subseteq Y \}.\) (The support of an n-tuple is the set of indices on which it is \(\neq 0.\)) Theorem: The union of the sets \({\mathcal C}(X,Y)\) with \(|X|,|Y|\leq 3\) coincides with the union of numerical ranges \(W(FG)\) where \(F,G\) run over the normal \(n\times n\) matrices \(F,G\) with eigenvalues \(f_1,\ldots f_n; g_1,\ldots, g_n.\) The proof rests on a formula for the trace class multiplier norm of a matrix of rank \(\leq 2\) derived by \textit{S. W. Drury} [Linear Algebra Appl. 280, No. 2-3, 217-227 (1998; Zbl 0935.15023)] and on use of von Neumann's minimax theorem. In an earlier section the author uses the results of that paper also to supplement his alternative approach to the maximal spectral distance problem solved by \textit{J. Holbrook, M. Omladic}, and \textit{P. Šemrl} [ibid. 249, 197-205 (1996; Zbl 0861.15019)] by a geometric description for the 2-dimensional case.
reviewed by: Alexander Kovačec
Mathematics Subject Classification ID: 15A60
Mathematics Subject Classification ID: 15A42
zbMATH DE Number: 1331350
zbMATH Keywords: Hadamard multiplier norm
zbMATH Keywords: maximal spectral distance
zbMATH Keywords: normal matrices
zbMATH Keywords: minimax theorem
zbMATH Keywords: numerical ranges
zbMATH Keywords: trace class multiplier norm
author: S. W. Drury
MaRDI profile type: Publication
cites work: Product of matrical discs∗
cites work: Trace class multipliers and spectral variation of normal matrices
cites work: Q5615632
cites work: Maximal spectral distance
cites work: Q4040931
cites work: On the Distance Between Normal Matrices
full work available at URL: s0024-3795(98)10224-0
OpenAlex ID: W2074798097
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description: scientific article; zbMATH DE number 1331350