Beitr\ EMIS ELibM Electronic Journals Beiträge zur Algebra und Geometrie
Contributions to Algebra and Geometry
Vol. 45, No. 2, pp. 435-446 (2004)

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Adjacency preserving mappings of rectangular matrices

Wen-ling Huang and Zhe-Xian Wan

Fachbereich Mathematik, Schwerpunkt GD, Universit{ä}t Hamburg, Bundesstr. 55, 20146 Hamburg, Germany, e-mail:; Academy of Mathematics and System Sciences, Chinese Academy of Sciences, 100080 Beijing, and Center for Combinatorics, Nankai University, 300071 Tienjin, China e-mail:

Abstract: Let $D$ be a division ring and let $m,n$ be integers $\ge 2$. Let $M_{m\times n}(D)$ be the space of $m\times n$ matrices. In the fundamental theorem of the geometry of rectangular matrices all bijective mappings $\vp$ of $M_{m\times n}(D)$ are determined such that both $\varphi$ and ${\varphi}^{-1}$ preserve adjacency. We show that if a bijective map $\varphi$ of $M_{m\times n}(D)$ preserves the adjacency then also ${\varphi}^{-1}$ preserves the adjacency. Thus the supposition that ${\varphi}p^{-1}$ preserves adjacency may be omitted in the fundamental theorem.

Keywords: Geometry of matrices, rectangular matrices, mappings preserving adjacency, distance preserving mappings

Classification (MSC2000): 15A99, 51D20

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Electronic version published on: 9 Sep 2004. This page was last modified: 4 May 2006.

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