Linear transformations preserving the Grassmannian over \(\mathcal M_n(\mathbb Z_+)\)
type: Item
instance of: scholarly article
title: Linear transformations preserving the Grassmannian over \(\mathcal M_n(\mathbb Z_+)\)
zbMATH Open document ID: 1067.15001
author: Alexander Guterman
author: Sang-Gu Lee
author: Seok-Zun Song
author: Leroy B. Beasley
published in: Linear Algebra and its Applications
publication date: 2005-01-11T00:00:00Z
review text: Let \({\mathbb Z}_+\) be the set of non-negative integers and \({\mathcal M}_n({\mathbb Z}_+)\) denote the set of \(n \times n\) matrices with coefficients from \({\mathbb Z}_+\). The semiring \({\mathbb Z}_+\) is an example of an antinegative semiring, i. e., one for which \(a+b=0\) implies that \(a=b=0\). The problem of determining the linear operators on the \(n \times n\) matrix algebra \({\mathcal M}_n({\mathbb F})\) over a field \({\mathbb F}\) that leave certain matrix relations invariant is widely studied. Here, a characterization of linear transformations on matrices with non-negative integer coefficients that preserve the matrix relation \(XYZ+ZYX=YXZ+ZXY\) is obtained. The polynomial \(XYZ+ZYX=YXZ+ZXY\) is known as a Grassmannian since it is identically zero if evaluated over the infinite dimensional Grassmann algebra [\textit{D. Krakowski} and \textit{A. Regev}, Trans. Am. Math. Soc. 181, 429--438 (1973; Zbl 0289.16015)].
Mathematics Subject Classification ID: 15A04
Mathematics Subject Classification ID: 15A30
Mathematics Subject Classification ID: 15A75
zbMATH DE Number: 2126340
zbMATH Keywords: linear preservers
zbMATH Keywords: Grassmannian
zbMATH Keywords: antinegative semiring
zbMATH Keywords: matrix algebra
zbMATH Keywords: linear transformations
zbMATH Keywords: Grassmann algebra
MaRDI profile type: Publication
full work available at URL: j.laa.2003.08.018
OpenAlex ID: W1966324003
Wikidata QID: Q57702708
cites work: The Polynomial Identities of the Grassman Algebra
cites work: Linear preserver problems: A brief introduction and some special techniques
cites work: Chapter 9: miscellaneous preserver problems
DOI: 10.1016/J.LAA.2003.08.018
Recommended article: Transformations of Grassmannians preserving the class of base subsets
Recommended article: Linear transformations preserving algebraic elements on \(\mathscr{M}_2(\mathbb{F})\) of degree 2
Recommended article: Orthogonality preserving transformations of Hilbert Grassmannians
Recommended article: Q4864989
Recommended article: Order preserving transformations of the Hilbert Grassmannian
Recommended article: Order preserving transformations of the Hilbert Grassmannian (note on the complex case)
Recommended article: Adjacency preserving transformations of Grassmann spaces
Recommended article: Commutativity-preserving and normal-preserving linear transformations on
Recommended article: Apartments preserving transformations of Grassmannians of infinite-dimensional vector spaces
Recommended article: Transformations of polar Grassmannians preserving certain intersecting relations
description: scientific article; zbMATH DE number 2126340