Totally geodesic submanifolds in the manifold SPD of symmetric positive-definite real matrices

Autores
Tumpach, Alice Barbara; Larotonda, Gabriel Andrés
Año de publicación
2024
Idioma
inglés
Tipo de recurso
artículo
Estado
versión publicada
Descripción
This paper is a self-contained exposition of the geometry of symmetric positive-definitereal n×n matrices SPD(n), including necessary and sufficent conditions for a submanifold N ⊂ SPD(n) to be totally geodesic for the affine-invariant Riemannian metric.A non-linear projection x → π(x) on a totally geodesic submanifold is defined.This projection has the minimizing property with respect to the Riemannian metric:it maps an arbitrary point x ∈ SPD(n) to the unique closest element π(x) in thetotally geodesic submanifold for the distance defined by the affine-invariant Riemannian metric. Decompositions of the space SPD(n) follow, as well as variants of thepolar decomposition of non-singular matrices known as Mostow’s decompositions.Applications to decompositions of covariant matrices are mentioned.
Fil: Tumpach, Alice Barbara. University Of Lille.; Francia. Technische Universitat Wien; Austria
Fil: Larotonda, Gabriel Andrés. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemática Alberto Calderón; Argentina
Materia
MATRIX FACTORIZATION
NONPOSITIVE CURVATURE
SYMMETRIC POSITIVE-DEFINITE REAL MATRIX
TOTALLY GEODESIC
Nivel de accesibilidad
acceso abierto
Condiciones de uso
https://creativecommons.org/licenses/by-nc-sa/2.5/ar/
Repositorio
CONICET Digital (CONICET)
Institución
Consejo Nacional de Investigaciones Científicas y Técnicas
OAI Identificador
oai:ri.conicet.gov.ar:11336/257863
Acceder
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network_name_str CONICET Digital (CONICET)
spelling Totally geodesic submanifolds in the manifold SPD of symmetric positive-definite real matricesTumpach, Alice BarbaraLarotonda, Gabriel AndrésMATRIX FACTORIZATIONNONPOSITIVE CURVATURESYMMETRIC POSITIVE-DEFINITE REAL MATRIXTOTALLY GEODESIChttps://purl.org/becyt/ford/1.1https://purl.org/becyt/ford/1This paper is a self-contained exposition of the geometry of symmetric positive-definitereal n×n matrices SPD(n), including necessary and sufficent conditions for a submanifold N ⊂ SPD(n) to be totally geodesic for the affine-invariant Riemannian metric.A non-linear projection x → π(x) on a totally geodesic submanifold is defined.This projection has the minimizing property with respect to the Riemannian metric:it maps an arbitrary point x ∈ SPD(n) to the unique closest element π(x) in thetotally geodesic submanifold for the distance defined by the affine-invariant Riemannian metric. Decompositions of the space SPD(n) follow, as well as variants of thepolar decomposition of non-singular matrices known as Mostow’s decompositions.Applications to decompositions of covariant matrices are mentioned.Fil: Tumpach, Alice Barbara. University Of Lille.; Francia. Technische Universitat Wien; AustriaFil: Larotonda, Gabriel Andrés. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemática Alberto Calderón; ArgentinaSpringer2024-11info:eu-repo/semantics/articleinfo:eu-repo/semantics/publishedVersionhttp://purl.org/coar/resource_type/c_6501info:ar-repo/semantics/articuloapplication/pdfapplication/pdfapplication/pdfhttp://hdl.handle.net/11336/257863Tumpach, Alice Barbara; Larotonda, Gabriel Andrés; Totally geodesic submanifolds in the manifold SPD of symmetric positive-definite real matrices; Springer; Information Geometry; 7; S2; 11-2024; 913-9422511-24812511-249XCONICET DigitalCONICETenginfo:eu-repo/semantics/altIdentifier/url/https://link.springer.com/article/10.1007/s41884-024-00146-zinfo:eu-repo/semantics/altIdentifier/doi/10.1007/s41884-024-00146-zinfo:eu-repo/semantics/altIdentifier/arxiv/https://arxiv.org/abs/2405.20784info:eu-repo/semantics/openAccesshttps://creativecommons.org/licenses/by-nc-sa/2.5/ar/reponame:CONICET Digital (CONICET)instname:Consejo Nacional de Investigaciones Científicas y Técnicas2025-10-22T11:12:02Zoai:ri.conicet.gov.ar:11336/257863instacron:CONICETInstitucionalhttp://ri.conicet.gov.ar/Organismo científico-tecnológicoNo correspondehttp://ri.conicet.gov.ar/oai/requestdasensio@conicet.gov.ar; lcarlino@conicet.gov.arArgentinaNo correspondeNo correspondeNo correspondeopendoar:34982025-10-22 11:12:02.632CONICET Digital (CONICET) - Consejo Nacional de Investigaciones Científicas y Técnicasfalse
dc.title.none.fl_str_mv Totally geodesic submanifolds in the manifold SPD of symmetric positive-definite real matrices
title Totally geodesic submanifolds in the manifold SPD of symmetric positive-definite real matrices
spellingShingle Totally geodesic submanifolds in the manifold SPD of symmetric positive-definite real matrices
Tumpach, Alice Barbara
MATRIX FACTORIZATION
NONPOSITIVE CURVATURE
SYMMETRIC POSITIVE-DEFINITE REAL MATRIX
TOTALLY GEODESIC
title_short Totally geodesic submanifolds in the manifold SPD of symmetric positive-definite real matrices
title_full Totally geodesic submanifolds in the manifold SPD of symmetric positive-definite real matrices
title_fullStr Totally geodesic submanifolds in the manifold SPD of symmetric positive-definite real matrices
title_full_unstemmed Totally geodesic submanifolds in the manifold SPD of symmetric positive-definite real matrices
title_sort Totally geodesic submanifolds in the manifold SPD of symmetric positive-definite real matrices
dc.creator.none.fl_str_mv Tumpach, Alice Barbara
Larotonda, Gabriel Andrés
author Tumpach, Alice Barbara
author_facet Tumpach, Alice Barbara
Larotonda, Gabriel Andrés
author_role author
author2 Larotonda, Gabriel Andrés
author2_role author
dc.subject.none.fl_str_mv MATRIX FACTORIZATION
NONPOSITIVE CURVATURE
SYMMETRIC POSITIVE-DEFINITE REAL MATRIX
TOTALLY GEODESIC
topic MATRIX FACTORIZATION
NONPOSITIVE CURVATURE
SYMMETRIC POSITIVE-DEFINITE REAL MATRIX
TOTALLY GEODESIC
purl_subject.fl_str_mv https://purl.org/becyt/ford/1.1
https://purl.org/becyt/ford/1
dc.description.none.fl_txt_mv This paper is a self-contained exposition of the geometry of symmetric positive-definitereal n×n matrices SPD(n), including necessary and sufficent conditions for a submanifold N ⊂ SPD(n) to be totally geodesic for the affine-invariant Riemannian metric.A non-linear projection x → π(x) on a totally geodesic submanifold is defined.This projection has the minimizing property with respect to the Riemannian metric:it maps an arbitrary point x ∈ SPD(n) to the unique closest element π(x) in thetotally geodesic submanifold for the distance defined by the affine-invariant Riemannian metric. Decompositions of the space SPD(n) follow, as well as variants of thepolar decomposition of non-singular matrices known as Mostow’s decompositions.Applications to decompositions of covariant matrices are mentioned.
Fil: Tumpach, Alice Barbara. University Of Lille.; Francia. Technische Universitat Wien; Austria
Fil: Larotonda, Gabriel Andrés. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemática Alberto Calderón; Argentina
description This paper is a self-contained exposition of the geometry of symmetric positive-definitereal n×n matrices SPD(n), including necessary and sufficent conditions for a submanifold N ⊂ SPD(n) to be totally geodesic for the affine-invariant Riemannian metric.A non-linear projection x → π(x) on a totally geodesic submanifold is defined.This projection has the minimizing property with respect to the Riemannian metric:it maps an arbitrary point x ∈ SPD(n) to the unique closest element π(x) in thetotally geodesic submanifold for the distance defined by the affine-invariant Riemannian metric. Decompositions of the space SPD(n) follow, as well as variants of thepolar decomposition of non-singular matrices known as Mostow’s decompositions.Applications to decompositions of covariant matrices are mentioned.
publishDate 2024
dc.date.none.fl_str_mv 2024-11
dc.type.none.fl_str_mv info:eu-repo/semantics/article
info:eu-repo/semantics/publishedVersion
http://purl.org/coar/resource_type/c_6501
info:ar-repo/semantics/articulo
format article
status_str publishedVersion
dc.identifier.none.fl_str_mv http://hdl.handle.net/11336/257863
Tumpach, Alice Barbara; Larotonda, Gabriel Andrés; Totally geodesic submanifolds in the manifold SPD of symmetric positive-definite real matrices; Springer; Information Geometry; 7; S2; 11-2024; 913-942
2511-2481
2511-249X
CONICET Digital
CONICET
url http://hdl.handle.net/11336/257863
identifier_str_mv Tumpach, Alice Barbara; Larotonda, Gabriel Andrés; Totally geodesic submanifolds in the manifold SPD of symmetric positive-definite real matrices; Springer; Information Geometry; 7; S2; 11-2024; 913-942
2511-2481
2511-249X
CONICET Digital
CONICET
dc.language.none.fl_str_mv eng
language eng
dc.relation.none.fl_str_mv info:eu-repo/semantics/altIdentifier/url/https://link.springer.com/article/10.1007/s41884-024-00146-z
info:eu-repo/semantics/altIdentifier/doi/10.1007/s41884-024-00146-z
info:eu-repo/semantics/altIdentifier/arxiv/https://arxiv.org/abs/2405.20784
dc.rights.none.fl_str_mv info:eu-repo/semantics/openAccess
https://creativecommons.org/licenses/by-nc-sa/2.5/ar/
eu_rights_str_mv openAccess
rights_invalid_str_mv https://creativecommons.org/licenses/by-nc-sa/2.5/ar/
dc.format.none.fl_str_mv application/pdf
application/pdf
application/pdf
dc.publisher.none.fl_str_mv Springer
publisher.none.fl_str_mv Springer
dc.source.none.fl_str_mv reponame:CONICET Digital (CONICET)
instname:Consejo Nacional de Investigaciones Científicas y Técnicas
reponame_str CONICET Digital (CONICET)
collection CONICET Digital (CONICET)
instname_str Consejo Nacional de Investigaciones Científicas y Técnicas
repository.name.fl_str_mv CONICET Digital (CONICET) - Consejo Nacional de Investigaciones Científicas y Técnicas
repository.mail.fl_str_mv dasensio@conicet.gov.ar; lcarlino@conicet.gov.ar
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score 12.8982525

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