The product field of values
type: Item
instance of: scholarly article
title: The product field of values
zbMATH Open document ID: 1272.15015
author: Daniel Corey
author: Charles R. Johnson
author: Ryan Kirk
author: Brian Lins
author: Ilya M. Spitkovsky
published in: Linear Algebra and its Applications
publication date: 2013-02-22T00:00:00Z
review text: The authors consider the following special generalization of the classical numerical range \[ P(A,B)=\{ \langle Ax,x\rangle \langle Bx, x\rangle \;:\;x\in {\mathbb C}^n,~ \|x\|=1\}, \] which they call the product field of values. Basic properties of the product field of values are established. The main results are Theorem 1: For \(A,B\in M_n(\mathbb{C})\), \(P(A,B)\) is ray convex and therefore simply convex. (Note that a connected set \(\Omega\) is ray convex if for each \(\alpha\notin\Omega\) there is a ray emanating from \(\alpha\) not intersecting \(\Omega\).) and Theorem 2: The interior of \(P(A,B)\) is empty if and only if one of the following conditions hold: (i) both \(A\) and \(B\) are scalar multiples of Hermitian matrices, (ii) both \(A\) and \(B\) are linear combinations of the identity matrix and some Hermitian matrix \(H\), and (iii) \(A\) and \(B^*\) are linearly dependent.
Mathematics Subject Classification ID: 15A60
Mathematics Subject Classification ID: 15B57
zbMATH DE Number: 6138404
zbMATH Keywords: field of values
zbMATH Keywords: numerical range
zbMATH Keywords: ray convex
zbMATH Keywords: Hermitian matrices
MaRDI profile type: Publication
full work available at URL: j.laa.2012.09.028
OpenAlex ID: W1989838911
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DOI: 10.1016/J.LAA.2012.09.028
description: scientific article; zbMATH DE number 6138404