A note on numerical range and product of matrices
type: Item
instance of: scholarly article
title: A note on numerical range and product of matrices
zbMATH Open document ID: 1264.15021
author: Che-Man Cheng
author: Yuan Gao
published in: Linear Algebra and its Applications
publication date: 2013-03-05T00:00:00Z
review text: Let \(A\) be a \(n\)-by-\(n\) matrix. We use \(\sigma(A)\) and \(W(A)\) for the spectrum and the numerical range of \(A\). A well-known result in matrix theory states that if \(A\) is positive semidefinite matrix, then \(\sigma(AB)\subseteq W(A)W(B)\) for all \(B\in M_n\). \textit{R. Alizadeh} [ibid. 437, No. 6, 1422--1425 (2012; Zbl 1259.15033)] proved the converse of this result. That is: Let \(A\in M_n\). Then \(\sigma(AB)\subseteq W(A)W(B)\) for all \(B\in M_n\) if and only if \(A\) is a scalar multiple of a positive semidefinite matrix. But if 0 is in the interior of \(W(A)\), then the presented proof does not work. In this paper, the authors prove this theorem again.
reviewed by: Maryam Khosravi
Mathematics Subject Classification ID: 15A60
Mathematics Subject Classification ID: 15B48
Mathematics Subject Classification ID: 15A18
zbMATH DE Number: 6141615
zbMATH Keywords: spectrum
zbMATH Keywords: numerical range
zbMATH Keywords: positive semidefinite matrix
MaRDI profile type: Publication
full work available at URL: j.laa.2012.12.031
OpenAlex ID: W2005205688
cites work: Numerical range and product of matrices
cites work: Q4040931
DOI: 10.1016/J.LAA.2012.12.031
description: scientific article; zbMATH DE number 6141615