A constructive proof of the Cartan-Dieudonné-Scherk theorem in the real or complex case
type: Item
instance of: scholarly article
title: A constructive proof of the Cartan-Dieudonné-Scherk theorem in the real or complex case
zbMATH Open document ID: 1210.15013
author: Chris Fuller
published in: Journal of Pure and Applied Algebra
publication date: 2011-02-18T00:00:00Z
review text: Let \(\mathbb{F}\) be a field with char\((\mathbb{F)}\neq2\), and \(S\) be an \(n\times n\) nonsingular symmetric matrix over \(\mathbb{F}\). An \(S\)-Householder matrix is an \(n\times n\) matrix of the form \(H=I-2uu^{T}S/(u^{T}Su)\) for a vector \(u\) such that \(u^{T}Su\neq0\). An \(n\times n\) matrix is \(S\)-orthogonal if \(Q^{T}SQ=S\) (so each \(S\)-Householder matrix is \(S\)-orthogonal). The Cartan-Dieudonné-Scherk (CDS) theorem states that every \(S\)-orthogonal matrix \(Q\) is a product of at most \(k\) \(S\)-Householder matrices where \(k=\text{rank}(Q-I)\) if \(S(Q-I)\) is not skew symmetric, and \(k=\text{rank}(Q-I)+2\) if \(S(Q-I)\) is skew symmetric. This value of \(k\) is best possible. There are many proofs of the CDS theorem, and some constructions which provide an explicit factorization into \(S\)-Householder matrices but using more than the minimal number of factors. In the case that \(\mathbb{F=R}\) or \(\mathbb{C}\) the author describes a constructive process to factor an \(S\)-orthogonal matrix \(Q\) into \(k\) \(S\)-Householder matrices where \(k\) is the CDS bound.
reviewed by: John D. Dixon
Mathematics Subject Classification ID: 15A23
Mathematics Subject Classification ID: 15B10
zbMATH DE Number: 5853399
zbMATH Keywords: Householder matrix
zbMATH Keywords: matrix factorization
zbMATH Keywords: orthogonal matrix
zbMATH Keywords: Cartan-Dieudonné-Scherk theorem
MaRDI profile type: Publication
full work available at URL: j.jpaa.2010.08.002
OpenAlex ID: W2073168665
cites work: The decomposition of an orthogonal transformation as a product of reflections
cites work: Q5775614
cites work: Q4520913
cites work: Q5711210
cites work: Q5689624
cites work: Matrix Analysis
cites work: \(\mathbb G\)-reflectors: Analogues of Householder transformations in scalar product spaces
cites work: Stabilized hyperbolic Householder transformations
cites work: On the Decomposition of Orthogonalities into Symmetries
cites work: Q5630276
cites work: Q4004234
cites work: Constructive ways for generating (generalized) real orthogonal matrices as products of (generalized) symmetries
DOI: 10.1016/J.JPAA.2010.08.002
description: scientific article; zbMATH DE number 5853399