Adjacency preserving bijection maps of Hermitian matrices over any division ring with an involution
type: Item
instance of: scholarly article
title: Adjacency preserving bijection maps of Hermitian matrices over any division ring with an involution
zbMATH Open document ID: 1106.14034
publication date: 2006-12-13T00:00:00Z
review text: Given a division ring \(D\) with an involution \(a\to \bar{a}\), let \( \mathcal{H}_{n}(D)\) be the space of \(n\times n\) Hermitian matrices, \(n\geq 2\), with respect to the involution. For \(H_{1},H_{2}\in \mathcal{H}_{n}(D)\) the arithmetic distance ad\((H_{1},H_{2})\) is defined to be the rank of \(H_{1}-H_{2},\) and two matrices are said to be adjacent if their arithmetic distance is 1. Let \(\varphi :\mathcal{H}_{n}(D) \to \mathcal{H}_{n}(D)\) be an adjacency-preserving map which is a bijection. An important theorem of the geometry of Hermitian matrices with involution says that, under certain conditions, \(\varphi (X)=\alpha ^{t} \bar{P}X^{\sigma }P+H_{0}\) for all \(X\in \mathcal{H}_{n}(D)\) where \(\alpha \) is a nonzero element of \(D\) such that \(\bar{\alpha}=\alpha ,\) \(P\in \text{GL}_{n}(D),\;\sigma :D\to D\) an automorphism commuting with the involution, and \(H_{0}\) a fixed element of \( \mathcal{H}_{n}(D).\) One of these ``certain conditions'' is that \(\varphi ^{-1}\) also preserve the adjacency. Let \(P\in \text{GL}_{n}(D).\) In this paper, it is shown that any adjacency-preserving bijective map \(\varphi \) on \(\mathcal{H}_{n}(D)\) such that \(\varphi (0)=0\) then there is a \(Q\in \text{GL}_{n}(D)\) such that \(\varphi (^{t}\bar{P} \mathcal{M}_{i}P)=\,^{t}\bar{Q}\mathcal{M}_{i}Q,\) where \(\mathcal{M} _{i}\) is the subset of \(\mathcal{H}_{n}(D)\) generated over \(F\) by the matrix consisting of a one in the \(i,i\)-position and zeros everywhere else. Additionally, the author investigates whether the condition on \(\varphi ^{-1} \) can be removed. It is proved that any adjacency-preserving bijection \( \varphi \) has the property that \(\varphi ^{-1}\) also preserves the adjacency. This answers an open question by Z. X. Wan. Furthermore, if \( \mathcal{H}_{n}(D)\neq \mathcal{S}_{3}(\mathbb{F}_{2})\) then \(\varphi \) in fact preserves arithmetic distance, where \( \mathcal{S}_{3}(\mathbb{F}_{2})\) is the set of \(3\times 3\) symmetric matric…
reviewed by: Alan Koch
Mathematics Subject Classification ID: 14L35
Mathematics Subject Classification ID: 51D20
Mathematics Subject Classification ID: 15B33
zbMATH DE Number: 5080003
zbMATH Keywords: division ring with involution
zbMATH Keywords: hermitian matrix
zbMATH Keywords: adjacency
zbMATH Keywords: geometry of matrices
author: Liping Huang
MaRDI profile type: Publication
full work available at URL: s10114-005-0770-7
OpenAlex ID: W2095705286
cites work: Geometry of symmetric matrices over any field with characteristic other than two
cites work: Q4292035
cites work: Q4306605
cites work: Q4879942
cites work: Q4835119
cites work: Q4879938
cites work: Geometry of \(2\times 2\) Hermitian matrices
cites work: Adjacency preserving mappings of symmetric and Hermitian matrices
cites work: The quaternion matrix equation ΣA i XB i =E
cites work: Additive maps on matrix algebras preserving invertibility
cites work: Additive maps preserving similarity of operators on Banach spaces
DOI: 10.1007/S10114-005-0770-7
published in: Acta Mathematica Sinica, English Series
Recommended article: Adjacency preserving bijective maps on triangular matrices over any division ring
Recommended article: Adjacency Preserving Maps on Hermitian Matrices
Recommended article: Adjacency preserving mappings of symmetric and Hermitian matrices
Recommended article: Adjacency preserving mappings of \(2 \times 2\) Hermitian matrices
Recommended article: Adjacency preservers on invertible Hermitian matrices. II.
Recommended article: Adjacency preservers on invertible Hermitian matrices. I.
Recommended article: A note on adjacency preservers on Hermitian matrices over finite fields
Recommended article: Maps preserving invertibility of matrices over semirings
Recommended article: Bijections preserving invertibility of differences of matrices on \(H_n\)
Recommended article: Orthogonality preserving bijective maps on finite dimensional projective spaces over division rings
description: scientific article; zbMATH DE number 5080003