On invariant subspaces of matrices: a new proof of a theorem of Halmos
type: Item
instance of: scholarly article
title: On invariant subspaces of matrices: a new proof of a theorem of Halmos
zbMATH Open document ID: 1206.15007
published in: Linear Algebra and its Applications
publication date: 2010-11-05T00:00:00Z
review text: \textit{P. R. Halmos} [Linear Algebra Appl. 4, 11--15 (1971; Zbl 0264.15001)], proved that if \(A\) is an \(n\times n\) matrix over the field of complex numbers and \(E\) is an \(A\)-invariant subspace, then there exist matrices \(B\) and \(C\) such that \(BA=AB\), \(CA=AC\), \(E\) is the kernel of \(B\) and \(E\) is the range of \(C\). In this paper the author gives an elementary elegant proof of this result and shows that there exist \(B\) and \(C\) that additionally satisfy \(BC=CB=0\). The author remarks that this result also holds for a matrix \(A\) with coefficients in any field \(K\).
reviewed by: Jaspal Singh Aujla
Mathematics Subject Classification ID: 15A18
Mathematics Subject Classification ID: 15A27
zbMATH DE Number: 5811039
zbMATH Keywords: invariant subspace
zbMATH Keywords: commutant
author: Ignat Domanov
MaRDI profile type: Publication
full work available at URL: j.laa.2010.07.033
OpenAlex ID: W2018298748
cites work: Invariant subspaces of a nilpotent operator on a Banach space.
cites work: Q3883021
cites work: Q4040150
cites work: On the spectral analysis of direct sums of Riemann-Liouville operators in Sobolev spaces of vector functions
cites work: On the orbit of invariant subspaces of linear operators in finite-dimensional spaces (new proof of a Halmos's result)
cites work: Eigenvectors and adjoints
cites work: Q3998992
cites work: On the similarity transformation between a matrix and its transpose
cites work: Q3935719
cites work: Q3704461
DOI: 10.1016/J.LAA.2010.07.033
description: scientific article; zbMATH DE number 5811039