Additive mappings on symmetric matrices
type: Item
instance of: scholarly article
title: Additive mappings on symmetric matrices
zbMATH Open document ID: 1108.15003
author: Bojan Kuzma
author: Marko Orel
published in: Linear Algebra and its Applications
publication date: 2006-11-15T00:00:00Z
review text: Let \(F_1\) and \(F_2\) be two fields and let \(S_n(F_1)\) and \(S_n(F_2)\) be the set of symmetric matrices over \(F_1\) and \(F_2\), respectively. A mapping \(\Phi: S_n(F_1)\to S_n(F_2)\) is called additive if \(\Phi(A+B)=\Phi(A)+\Phi(B)\). It is said that \(\Phi\) doesn't increase rank-one if \(rk(\Phi(A))\leq1\) whenever \(rk(A)=1\). \textit{M. H. Lim} [Linear Algebra Appl. 402, 263--271 (2005; Zbl 1084.15008)] has characterized the class of rank-\(1\) nonincreasing additive mappings on the symmetric matrices for the fields of characteristic \(\neq{2}\) and \(3\). The authors in the paper under review characterize all such additive mappings in characteristic \(2\) and \(3\). They use bases for matrix spaces and prove their results case by case. Although influenced by Lim [loc. cit.], they fill in the missing gaps in the mentioned paper.
reviewed by: Manouchehr Misaghian
Mathematics Subject Classification ID: 15A04
Mathematics Subject Classification ID: 15A03
Mathematics Subject Classification ID: 15B57
zbMATH DE Number: 5072868
zbMATH Keywords: rank
zbMATH Keywords: additive preserver
zbMATH Keywords: symmetric matrix
MaRDI profile type: Publication
full work available at URL: j.laa.2006.02.010
OpenAlex ID: W1965284066
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DOI: 10.1016/J.LAA.2006.02.010
description: scientific article; zbMATH DE number 5072868