Diameter preserving surjection on alternate matrices
type: Item
instance of: scholarly article
title: Diameter preserving surjection on alternate matrices
zbMATH Open document ID: 1187.15030
publication date: 2009-12-11T00:00:00Z
review text: Let \(F^{m\times n}\) be the set of all \(m\times n\) matrices over a field \(F\). A matrix \(A\in F^{m\times m}\) is called an alternate matrix if \(^tA=-A\) with zero diagonal entries, where \(^tA\) is the transpose of \(A\). The set of all \(m\times m\) alternate matrices over \(F\) is denoted by \( \mathcal {K}_m(F)\). Note that a matrix \(A\in F^{m\times m}\) is an alternate matrix if and only if \(^txAx=0\) for all \(x\in F^{m\times 1}\). If the characteristic of \(F\) is not 2, \(A\) is alternate if and only if \(A\) is skew-symmetric. The arithmetic distance between \(A, B\in \mathcal {K}_m(F)\) is \(d(A, B)= \text{rank}(A-B)\). \(A\) and \(B\) are said to be adjacent if \(d(A, B)=2\) and denoted by \(A\sim B\). The diameter of \(\mathcal {K}_m(F)\) is defined as \(d_{\max}=\max \{ d(A, B): A, B\in \mathcal {K}_m(F)\} \). Let \(\varphi : \mathcal {K}_m(F)\to \mathcal {K}_m(F)\) be a map. Then \(\varphi \) is said to be (1) an adjacency preserving map if \[ A \sim B \Rightarrow \varphi (A) \sim \varphi (B); \] (2) an adjacency preserving map in both directions if \[ A\sim B \Leftrightarrow \varphi (A) \sim \varphi (B); \] (3) an arithmetic distance preserving map if \[ d(\varphi (A), \varphi (B))=d(A, B); \] (4) a diameter preserving map if \[ d(A, B)=d_{\max } \Rightarrow d(\varphi (A), \varphi (B))=d_{\max }; \] (5) a diameter preserving map in both directions if \[ d(A, B)=d_{\max } \Leftrightarrow d(\varphi (A), \varphi (B))=d_{\max }; \] (6) a preserver of distance-additivity if \[ d(A, B)=d(A, C)+d(C, B) \Rightarrow d(\varphi (A), \varphi (B))=d(\varphi (A), \varphi (C))+d(\varphi (C), \varphi (B)), \] for all \(A, B, C\in \mathcal {K}_m(F)\). The aim of this paper is to investigate these preservers on alternate matrices. The following result is obtained: Let \(F\) be a field with \(|F|\geq 3\) and \(m\geq 4\) be an integer. If \(\varphi \) is a map from \(\mathcal {K}_m(F)\) into itself, then the following statements are equivalent: (i) \(\varphi \) is…
reviewed by: Wu Jing
Mathematics Subject Classification ID: 15A86
Mathematics Subject Classification ID: 51D20
zbMATH DE Number: 5645776
zbMATH Keywords: geometry of matrices
zbMATH Keywords: alternate matrix
zbMATH Keywords: arithmetic distance
zbMATH Keywords: adjacency
zbMATH Keywords: diameter
zbMATH Keywords: preserver problems
zbMATH Keywords: skew-symmetric
zbMATH Keywords: rank
author: Liping Huang
MaRDI profile type: Publication
full work available at URL: s10114-009-7697-3
OpenAlex ID: W2137499988
cites work: Q5545582
cites work: Adjacency preserving bijection maps of Hermitian matrices over any division ring with an involution
cites work: Adjacency preserving mappings of symmetric and Hermitian matrices
cites work: Adjacency preserving mappings of rectangular matrices
cites work: Geometry of block triangular matrices over a division ring
cites work: On Hua's fundamental theorem of the geometry of rectangular matrices
cites work: Additive map preserving rank 2 on alternate matrices
cites work: Additive preservers of rank-additivity on the spaces of symmetric and alternate matrices.
cites work: Additive preservers of rank on alternate matrix spaces over fields and applications
cites work: Linear/additive preservers of rank 2 on spaces of alternate matrices over fields
cites work: From geometry to invertibility preservers
DOI: 10.1007/S10114-009-7697-3
published in: Acta Mathematica Sinica, English Series
Recommended article: Diameter preserving surjections in the geometry of matrices
Recommended article: Bounded distance preserving surjections in the geometry of matrices
Recommended article: Surjective maps on matrices preserving the local spectral radius distance
Recommended article: ON DIAMETER PRESERVING LINEAR MAPS
Recommended article: Bounded distance preserving surjections in the projective geometry of matrices
Recommended article: Linear bijections which preserve the diameter of vector-valued maps
Recommended article: Diameter preserving linear bijections of \(C(X)\)
Recommended article: Diameter preserving linear maps and isometries
Recommended article: Additive rank-one preserving surjections on symmetric matrix spaces
Recommended article: Q4258704
description: scientific article; zbMATH DE number 5645776