On linear preservers of normal maps
type: Item
instance of: scholarly article
title: On linear preservers of normal maps
zbMATH Open document ID: 1139.06010
author: M. Niezgoda
published in: Linear Algebra and its Applications
publication date: 2007-09-13T00:00:00Z
review text: For a closed subgroup \(G\) of an orthogonal group on a real finite-dimensional vector space \(V\), an order related to \(G\) is defined by \(y\leq_G x\) if \(y\) is in the convex hull of \(Gx=\{gx:g \in G \}\). The order is said to be a group induced cone (GIC) if there exists a closed convex cone \(D\) with \(D\cap Gx\) non-empty for all \(x\in V\) and \(\langle x,gy\rangle\leq\langle x,y\rangle\) \((x,y\in D\), \(g\in G)\). The map assigning \(x\) the element \(D\cap Gx\) (one element) is called the normal map. For example, the eigenvalue map on the Hermitian matrices (\(n\times n\)) can be viewed as a normal map. For two linear spaces \(V\) and \( W\) together with associated groups and cones, a map \(K\) from \(V\) to \(W\) is called a preserver of normal maps if the normal map of \(Kx\) is equal to \(K\) of the normal map of \(x\). The paper studies preservers of normal maps and properties maintained under the action of these preservers. It is shown, for example, that preservers of normal maps are positive operators preserving the order (induced by the subgroups) with the image of the cone of \(V\) a subset of the cone of \(W\). Numerous examples are provided.
reviewed by: William A. Feldman
Mathematics Subject Classification ID: 06F20
Mathematics Subject Classification ID: 20F55
Mathematics Subject Classification ID: 15A30
Mathematics Subject Classification ID: 15A18
zbMATH DE Number: 5190767
zbMATH Keywords: group majorization
zbMATH Keywords: GIC ordering
zbMATH Keywords: finite reflection group
zbMATH Keywords: linear preserver
zbMATH Keywords: singular value
zbMATH Keywords: eigenvalue
MaRDI profile type: Publication
full work available at URL: j.laa.2007.04.007
OpenAlex ID: W2171292847
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DOI: 10.1016/J.LAA.2007.04.007
description: scientific article; zbMATH DE number 5190767