Adjacency preserving mappings on real symmetric matrices
type: Item
instance of: scholarly article
zbMATH Open document ID: 1262.15033
publication date: 2012-07-16T00:00:00Z
full work available at URL: https://arxiv.org/abs/0711.2413
full work available at URL: http://hrcak.srce.hr/index.php
Mathematics Subject Classification ID: 15A86
Mathematics Subject Classification ID: 15A03
Mathematics Subject Classification ID: 15A30
Mathematics Subject Classification ID: 15B57
zbMATH DE Number: 6057204
zbMATH Keywords: rank
zbMATH Keywords: geometry of matrices
zbMATH Keywords: real symmetric matrix
zbMATH Keywords: adjacency preserving map
author: Peter Legiša
MaRDI profile type: Publication
arXiv classification: math.RA
arXiv classification: math.MG
arXiv ID: 0711.2413
title: Adjacency preserving mappings on real symmetric matrices
published in: Mathematical Communications
review text: The author characterizes adjacency preserving maps \(\varphi\) between spaces of real symmetric matrices \(S_n({\mathbb R})\) and \(S_m({\mathbb R})\) under the additional assumption that \(\varphi(0)=0\). It is shown that either \(\varphi\) maps the whole space \(S_n({\mathbb R})\) into \({\mathbb R}B\), where \(B\) is a symmetric matrix of rank 1, or there exist \(c\in\{\pm 1\}\) and an invertible matrix \(R\in M_m({\mathbb R})\) such that for all \(A\in S_n({\mathbb R})\) it holds that \(\varphi (A)=cR (A \oplus 0_{m-n}) R^t\), here \(m\geq n\), and if \(m=n\) then there are no zero blocks on the righthand side.
reviewed by: Alexander Guterman
description: scientific article; zbMATH DE number 6057204