Adjacency preserving mappings of symmetric and Hermitian matrices
type: Item
instance of: scholarly article
title: Adjacency preserving mappings of symmetric and Hermitian matrices
zbMATH Open document ID: 1061.15021
author: Roland Höfer
author: Wen-ling Huang
author: Zhe-Xian Wan
published in: Aequationes Mathematicae
publication date: 2004-11-26T00:00:00Z
review text: The space \(\mathcal{H}_n(D)\), where \(n\) is a natural number and \(D\) is a division ring, denotes the graph on the \((n \times n)\)-Hermitian matrices over \(D\) in which two matrices \(A\) and \(B\) are adjacent if the rank of the matrix \(A-B\) is \(1\). The authors prove the following theorem: Let \(n \geq 2\), let \(D\) be a division ring which admits an involution \(\;\bar{}\;\), and let \(F\) be the set of fixed elements of \(\;\bar{}\;\). If \(\;\bar{}\;\) is the identity, assume furthermore that the characteristic of the field \(D = F\) is distinct from \(2\). If \(\;\bar{}\;\) is not the identity, assume that \(F\) is a central subfield of \(D\). If a bijective map \(\phi\) from \(\mathcal{H}_n(D)\) onto itself preserves adjacency in \(\mathcal{H}_n(D)\), then so does its inverse \(\phi^{-1}\).
Mathematics Subject Classification ID: 15B57
Mathematics Subject Classification ID: 15A30
Mathematics Subject Classification ID: 51D20
Mathematics Subject Classification ID: 05C50
zbMATH DE Number: 2119230
zbMATH Keywords: adjacency preserving mappings
zbMATH Keywords: symmetric matrices
zbMATH Keywords: Hermitian matrices
zbMATH Keywords: division ring
zbMATH Keywords: rank
zbMATH Keywords: involution
MaRDI profile type: Publication
full work available at URL: s00010-003-2689-3
OpenAlex ID: W2158388199
DOI: 10.1007/S00010-003-2689-3
description: scientific article; zbMATH DE number 2119230