A vector-bundle version of a theorem of V. Doležal
type: Item
instance of: scholarly article
title: A vector-bundle version of a theorem of V. Doležal
zbMATH Open document ID: 1065.15030
author: Kevin A. Grasse
published in: Linear Algebra and its Applications
publication date: 2004-11-26T00:00:00Z
review text: A well known theorem of \textit{V. Doležal} [Časopis Mat. 89, 466--468 (1964; Zbl 0187.37103)] states that given a square matrix \(A(t)\) that is a \(C^k\) function of the scalar parameter \(t\in\mathbb{R}\) and whose rank is constant as the parameter varies, one can span the range and kernel of \(A(t)\) by linearly independent vectors that are also \(C^k\) functions of \(t\). The paper explores some generalizations of Doležal's theorem in which the parameter space \(\mathbb{R}\) is replaced by a topological space when \(k= 0\) or a differentiable manifold when \(k\geq 1\). The author shows that for a broad class of parameter spaces one can associate two vector bundles to a constant-rank, \(C^k\) parameterized matrix function and that Dolžal's theorem will continue to hold if and only if both of these vector bundles are trivial. In particular, this result generalizes Doležal's theorem to the case where the parameter space is contractible.
Mathematics Subject Classification ID: 15A54
zbMATH DE Number: 2119127
zbMATH Keywords: Doležal's theorem
zbMATH Keywords: Matrices depending on a parameter
zbMATH Keywords: Vector bundles
zbMATH Keywords: Contractible spaces
zbMATH Keywords: Block singular value decomposition
zbMATH Keywords: range
zbMATH Keywords: kernel
zbMATH Keywords: topological space
zbMATH Keywords: differentiable manifold
zbMATH Keywords: matrix function
reviewed by: Rodica Covaci
MaRDI profile type: Publication
full work available at URL: j.laa.2004.05.014
OpenAlex ID: W2073390367
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DOI: 10.1016/J.LAA.2004.05.014
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description: scientific article; zbMATH DE number 2119127