Additive preservers of rank on alternate matrix spaces over fields and applications
type: Item
instance of: scholarly article
title: Additive preservers of rank on alternate matrix spaces over fields and applications
zbMATH Open document ID: 1088.15005
author: Xian Zhang
published in: Linear Algebra and its Applications
publication date: 2005-05-04T00:00:00Z
review text: Let us denote by \(F\) any field and by \(A^t\) the transpose of any matrix \(A\) with entries in \(F\). A square matrix \(A\) is said to be \(alternate\) if \(A^t=-A\) and all diagonal elements are zeros. These matrices play a considerable role in the theory of quadratic forms and classical groups, and therefore the endomorphisms preserving their properties take on great interest. The author determines the general form of all additive preservers of rank, rank-additivity and rank-subtractivity. Assume that \(K_n (F )\) is the set of all \(n \times n,n\geq 4\) alternate matrices over \(F\) ; then \((K_n (F) , + , \cdot ) \) is a non-associative ring under the usual addition ``\(+\)'' and the multiplication ``\(\cdot\)'' defined in the following way: \(X \cdot Y=XYX \) for all \( X , Y \in K_n(F)\). The author characterizes all ring endomorphisms of \((K_n (F),+ , \cdot)\), i.e. all the additive maps \(\phi\) such that \(\phi (X \cdot Y ) = \phi (X) \cdot \phi (Y)\) for any \(X , Y \in K_n (F)\).
reviewed by: Vito Abatangelo
Mathematics Subject Classification ID: 15A04
Mathematics Subject Classification ID: 15A03
zbMATH DE Number: 2165365
zbMATH Keywords: alternate matrices
zbMATH Keywords: rank-additivity
zbMATH Keywords: rank-subtractivity
zbMATH Keywords: non-associative ring
zbMATH Keywords: additive preserver
MaRDI profile type: Publication
full work available at URL: j.laa.2004.11.006
OpenAlex ID: W1978599007
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DOI: 10.1016/J.LAA.2004.11.006
description: scientific article; zbMATH DE number 2165365