Hua's fundamental theorem of the geometry of matrices.
type: Item
instance of: scholarly article
title: Hua's fundamental theorem of the geometry of matrices.
zbMATH Open document ID: 1085.15019
author: Peter Šemrl
published in: Journal of Algebra
publication date: 2004-03-14T00:00:00Z
review text: Let \(M_n(F)\) be the space of all \(n\times n\) matrices over the field \(F\), \(n\geq 2\). Two matrices, \(A, B\in M_n(F)\) are adjacent if \(\text{rank}(A-B)= 1\). \textit{L. K. Hua}'s fundamental theorem of the geometry of square matrices [A theorem on matrices over a field and its applications, Acta Math. Sinica 1, 109--163 (1951)] characterizes bijective maps on \(M_n(F)\) that preserve adjacency in both directions. In this paper the author treats an open problem whether the result of Hua holds true under the weaker assumption of preserving the adjacency in one direction only. The author answers this question in the affirmative in the case that every nonzero homomorphism \(f:F\to F\) is surjective. In order to prove this result the author has to improve \textit{P. G. Ovchinnikov}'s characterization of automorphisms of the poset of idempotent matrices [J. Funct. Anal. 115, No. 1, 184--189 (1993; Zbl 0806.46069)]. The author uses a method to improve the infinite-dimensional part of Ovchinnikov's result and characterizes injective continuous order-preserving maps on complex idempotent matrices. The author concludes the paper by describing some applications of the results.
Mathematics Subject Classification ID: 15A30
zbMATH DE Number: 2055654
zbMATH Keywords: Hua's fundamental theorem
zbMATH Keywords: bijective maps
zbMATH Keywords: adjacency
zbMATH Keywords: surjective
zbMATH Keywords: idempotent matrices
reviewed by: Yueh-er Kuo
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2003.07.019
OpenAlex ID: W1989229665
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DOI: 10.1016/J.JALGEBRA.2003.07.019
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description: scientific article; zbMATH DE number 2055654