Structural matrix algebras and their lattices of invariant subspaces
type: Item
instance of: scholarly article
title: Structural matrix algebras and their lattices of invariant subspaces
zbMATH Open document ID: 1068.15014
author: Mustafa Akkurt
author: George Phillip Barker
author: Marcel Wild
published in: Linear Algebra and its Applications
publication date: 2005-02-23T00:00:00Z
review text: Let \(M_{n}(F)\) be the \(F\)-algebra of all \(n\times n\) matrices over a field \(F\). For any transitive and reflexive binary relation \(\rho\) on \(\left\{ 1,2,\dots,n\right\} \), the set \(M_{n}(F,\rho)\) of all matrices \(\left[ a_{ij}\right] \) in \(M_{n}(F)\) for which \(a_{ij}=0\) for all \((i,j)\notin\rho\) is a subalgebra of \(M_{n}(F)\) called a ``structural matrix algebra''. Structural matrix algebras have been studied (under various names) in papers ranging from \textit{R. M. Thrall} [Can. J. Math. 4, 227--239 (1952; Zbl 0048.02304)] to \textit{J. H. Meyer} and \textit{L. van Wyk} [Chin. J. Math. 24, 251--264 (1996; Zbl 0869.16018)]. The authors give new proofs of some of the earlier results, including the fact that the lattice of \(M_{n}(F,\rho)\)-invariant subspaces of \(F^{n}\) is always distributive, and providing a suitable converse. They also examine the situation where a subalgebra \(\mathcal{R}\) of \(M_{n}(F)\) is isomorphic to some structural matrix algebra \(M_{m}(F,\rho)\) with \(m<n\). In the latter case the lattice of \(\mathcal{R}\)-invariant subspaces of \(F^{n}\) is not necessarily distributive, but it contains a specified sublattice which is distributive.
reviewed by: John D. Dixon
Mathematics Subject Classification ID: 15A30
Mathematics Subject Classification ID: 15A18
Mathematics Subject Classification ID: 16S50
Mathematics Subject Classification ID: 06D05
zbMATH DE Number: 2137770
zbMATH Keywords: invariant subspace
zbMATH Keywords: distributive lattice
zbMATH Keywords: structural matrix algebra
MaRDI profile type: Publication
full work available at URL: j.laa.2004.05.009
OpenAlex ID: W2114588098
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DOI: 10.1016/J.LAA.2004.05.009
description: scientific article; zbMATH DE number 2137770