An algorithm for the Cartan-Dieudonné theorem on generalized scalar product spaces

M. A. Rodríguez-Andrade; G. Aragón-González; J. L. Aragón; Luis Verde-Star

doi:10.1016/j.laa.2010.11.005

An algorithm for the Cartan-Dieudonné theorem on generalized scalar product spaces

M. A. Rodríguez-Andrade, G. Aragón-González, J. L. Aragón, Luis Verde-Star

Escuela Superior de Física y Matemáticas (ESFM)

Research output: Contribution to journal › Article › peer-review

12 Scopus citations

Abstract

We present an algorithmic proof of the Cartan-Dieudonné theorem on generalized real scalar product spaces with arbitrary signature. We use Clifford algebras to compute the factorization of a given transformation as a product of reflections with respect to hyperplanes. The relationship with the Cartan-Dieudonné-Scherk theorem is also discussed in relation to the minimum number of reflections required to decompose a given orthogonal transformation.

Original language	English
Pages (from-to)	1238-1254
Number of pages	17
Journal	Linear Algebra and Its Applications
Volume	434
Issue number	5
DOIs	https://doi.org/10.1016/j.laa.2010.11.005
State	Published - 1 Mar 2011

Keywords

Cartan-Dieudonné
Clifford algebras
Householder transformations
Orthogonal group
Orthogonal matrices

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10.1016/j.laa.2010.11.005

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title = "An algorithm for the Cartan-Dieudonn{\'e} theorem on generalized scalar product spaces",

abstract = "We present an algorithmic proof of the Cartan-Dieudonn{\'e} theorem on generalized real scalar product spaces with arbitrary signature. We use Clifford algebras to compute the factorization of a given transformation as a product of reflections with respect to hyperplanes. The relationship with the Cartan-Dieudonn{\'e}-Scherk theorem is also discussed in relation to the minimum number of reflections required to decompose a given orthogonal transformation.",

keywords = "Cartan-Dieudonn{\'e}, Clifford algebras, Householder transformations, Orthogonal group, Orthogonal matrices",

author = "Rodr{\'i}guez-Andrade, {M. A.} and G. Arag{\'o}n-Gonz{\'a}lez and Arag{\'o}n, {J. L.} and Luis Verde-Star",

note = "Funding Information: MAR would like to thank for financial support from COFAA-IPN. GAG was supported by the Program for the Professional Development in Automation, through the grant from the Universidad Aut{\'o}noma Metropolitana. JLA would like to thank DGAPA-UNAM (grant IN100310-3) and CONACyT (grant 50368 and 79641) for financial support.",

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AU - Rodríguez-Andrade, M. A.

AU - Aragón-González, G.

AU - Aragón, J. L.

AU - Verde-Star, Luis

N1 - Funding Information: MAR would like to thank for financial support from COFAA-IPN. GAG was supported by the Program for the Professional Development in Automation, through the grant from the Universidad Autónoma Metropolitana. JLA would like to thank DGAPA-UNAM (grant IN100310-3) and CONACyT (grant 50368 and 79641) for financial support.

PY - 2011/3/1

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N2 - We present an algorithmic proof of the Cartan-Dieudonné theorem on generalized real scalar product spaces with arbitrary signature. We use Clifford algebras to compute the factorization of a given transformation as a product of reflections with respect to hyperplanes. The relationship with the Cartan-Dieudonné-Scherk theorem is also discussed in relation to the minimum number of reflections required to decompose a given orthogonal transformation.

AB - We present an algorithmic proof of the Cartan-Dieudonné theorem on generalized real scalar product spaces with arbitrary signature. We use Clifford algebras to compute the factorization of a given transformation as a product of reflections with respect to hyperplanes. The relationship with the Cartan-Dieudonné-Scherk theorem is also discussed in relation to the minimum number of reflections required to decompose a given orthogonal transformation.

KW - Cartan-Dieudonné

KW - Clifford algebras

KW - Householder transformations

KW - Orthogonal group

KW - Orthogonal matrices

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DO - 10.1016/j.laa.2010.11.005

M3 - Artículo

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SP - 1238

EP - 1254

JO - Linear Algebra and Its Applications

JF - Linear Algebra and Its Applications

IS - 5

ER -

An algorithm for the Cartan-Dieudonné theorem on generalized scalar product spaces

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Keywords

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