The left invariant metric in the general linear group

Autores
Andruchow, Esteban; Larotonda, Gabriel Andrés; Recht, Lázaro; Varela, Alejandro
Año de publicación
2014
Idioma
inglés
Tipo de recurso
artículo
Estado
versión publicada
Descripción
Left invariant metrics induced by the p-norms of the trace in the matrix algebra are studied on the general linear group. By means of the Euler–Lagrange equations, existence and uniqueness of extremal paths for the length functional are established, and regularity properties of these extremal paths are obtained. Minimizing paths in the group are shown to have a velocity with constant singular values and multiplicity. In several special cases, these geodesic paths are computed explicitly. In particular the Riemannian geodesics, corresponding to the case p=2, are characterized as the product of two one-parameter groups. It is also shown that geodesics are one-parameter groups if and only if the initial velocity is a normal matrix. These results are further extended to the context of compact operators with p-summable spectrum, where a differential equation for the spectral projections of the velocity vector of an extremal path is obtained.
Fil: Andruchow, Esteban. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemática; Argentina. Universidad Nacional de General Sarmiento; Argentina
Fil: Larotonda, Gabriel Andrés. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemática; Argentina. Universidad Nacional de General Sarmiento; Argentina
Fil: Recht, Lázaro. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemática; Argentina. Universidad Simón Bolívar; Venezuela
Fil: Varela, Alejandro. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemática; Argentina. Universidad Nacional de General Sarmiento; Argentina
Materia
General Linear Group
Left-Invariant Metric
P-Norm
Finsler Metric
Nivel de accesibilidad
acceso abierto
Condiciones de uso
https://creativecommons.org/licenses/by-nc-nd/2.5/ar/
Repositorio
CONICET Digital (CONICET)
Institución
Consejo Nacional de Investigaciones Científicas y Técnicas
OAI Identificador
oai:ri.conicet.gov.ar:11336/12163
Acceder
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oai_identifier_str oai:ri.conicet.gov.ar:11336/12163
network_acronym_str CONICETDig
repository_id_str 3498
network_name_str CONICET Digital (CONICET)
spelling The left invariant metric in the general linear groupAndruchow, EstebanLarotonda, Gabriel AndrésRecht, LázaroVarela, AlejandroGeneral Linear GroupLeft-Invariant MetricP-NormFinsler Metrichttps://purl.org/becyt/ford/1.1https://purl.org/becyt/ford/1Left invariant metrics induced by the p-norms of the trace in the matrix algebra are studied on the general linear group. By means of the Euler–Lagrange equations, existence and uniqueness of extremal paths for the length functional are established, and regularity properties of these extremal paths are obtained. Minimizing paths in the group are shown to have a velocity with constant singular values and multiplicity. In several special cases, these geodesic paths are computed explicitly. In particular the Riemannian geodesics, corresponding to the case p=2, are characterized as the product of two one-parameter groups. It is also shown that geodesics are one-parameter groups if and only if the initial velocity is a normal matrix. These results are further extended to the context of compact operators with p-summable spectrum, where a differential equation for the spectral projections of the velocity vector of an extremal path is obtained.Fil: Andruchow, Esteban. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemática; Argentina. Universidad Nacional de General Sarmiento; ArgentinaFil: Larotonda, Gabriel Andrés. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemática; Argentina. Universidad Nacional de General Sarmiento; ArgentinaFil: Recht, Lázaro. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemática; Argentina. Universidad Simón Bolívar; VenezuelaFil: Varela, Alejandro. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemática; Argentina. Universidad Nacional de General Sarmiento; ArgentinaElsevier Science2014-12info:eu-repo/semantics/articleinfo:eu-repo/semantics/publishedVersionhttp://purl.org/coar/resource_type/c_6501info:ar-repo/semantics/articuloapplication/pdfapplication/pdfapplication/pdfapplication/pdfapplication/pdfhttp://hdl.handle.net/11336/12163Andruchow, Esteban; Larotonda, Gabriel Andrés; Recht, Lázaro; Varela, Alejandro; The left invariant metric in the general linear group; Elsevier Science; Journal Of Geometry And Physics; 86; 12-2014; 241-2570393-0440enginfo:eu-repo/semantics/altIdentifier/url/http://www.sciencedirect.com/science/article/pii/S0393044014001879info:eu-repo/semantics/altIdentifier/doi/10.1016/j.geomphys.2014.08.009info:eu-repo/semantics/altIdentifier/url/https://arxiv.org/abs/1109.0520info:eu-repo/semantics/openAccesshttps://creativecommons.org/licenses/by-nc-nd/2.5/ar/reponame:CONICET Digital (CONICET)instname:Consejo Nacional de Investigaciones Científicas y Técnicas2025-09-29T10:06:57Zoai:ri.conicet.gov.ar:11336/12163instacron: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:34982025-09-29 10:06:57.884CONICET Digital (CONICET) - Consejo Nacional de Investigaciones Científicas y Técnicasfalse
dc.title.none.fl_str_mv The left invariant metric in the general linear group
title The left invariant metric in the general linear group
spellingShingle The left invariant metric in the general linear group
Andruchow, Esteban
General Linear Group
Left-Invariant Metric
P-Norm
Finsler Metric
title_short The left invariant metric in the general linear group
title_full The left invariant metric in the general linear group
title_fullStr The left invariant metric in the general linear group
title_full_unstemmed The left invariant metric in the general linear group
title_sort The left invariant metric in the general linear group
dc.creator.none.fl_str_mv Andruchow, Esteban
Larotonda, Gabriel Andrés
Recht, Lázaro
Varela, Alejandro
author Andruchow, Esteban
author_facet Andruchow, Esteban
Larotonda, Gabriel Andrés
Recht, Lázaro
Varela, Alejandro
author_role author
author2 Larotonda, Gabriel Andrés
Recht, Lázaro
Varela, Alejandro
author2_role author
author
author
dc.subject.none.fl_str_mv General Linear Group
Left-Invariant Metric
P-Norm
Finsler Metric
topic General Linear Group
Left-Invariant Metric
P-Norm
Finsler Metric
purl_subject.fl_str_mv https://purl.org/becyt/ford/1.1
https://purl.org/becyt/ford/1
dc.description.none.fl_txt_mv Left invariant metrics induced by the p-norms of the trace in the matrix algebra are studied on the general linear group. By means of the Euler–Lagrange equations, existence and uniqueness of extremal paths for the length functional are established, and regularity properties of these extremal paths are obtained. Minimizing paths in the group are shown to have a velocity with constant singular values and multiplicity. In several special cases, these geodesic paths are computed explicitly. In particular the Riemannian geodesics, corresponding to the case p=2, are characterized as the product of two one-parameter groups. It is also shown that geodesics are one-parameter groups if and only if the initial velocity is a normal matrix. These results are further extended to the context of compact operators with p-summable spectrum, where a differential equation for the spectral projections of the velocity vector of an extremal path is obtained.
Fil: Andruchow, Esteban. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemática; Argentina. Universidad Nacional de General Sarmiento; Argentina
Fil: Larotonda, Gabriel Andrés. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemática; Argentina. Universidad Nacional de General Sarmiento; Argentina
Fil: Recht, Lázaro. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemática; Argentina. Universidad Simón Bolívar; Venezuela
Fil: Varela, Alejandro. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemática; Argentina. Universidad Nacional de General Sarmiento; Argentina
description Left invariant metrics induced by the p-norms of the trace in the matrix algebra are studied on the general linear group. By means of the Euler–Lagrange equations, existence and uniqueness of extremal paths for the length functional are established, and regularity properties of these extremal paths are obtained. Minimizing paths in the group are shown to have a velocity with constant singular values and multiplicity. In several special cases, these geodesic paths are computed explicitly. In particular the Riemannian geodesics, corresponding to the case p=2, are characterized as the product of two one-parameter groups. It is also shown that geodesics are one-parameter groups if and only if the initial velocity is a normal matrix. These results are further extended to the context of compact operators with p-summable spectrum, where a differential equation for the spectral projections of the velocity vector of an extremal path is obtained.
publishDate 2014
dc.date.none.fl_str_mv 2014-12
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/11336/12163
Andruchow, Esteban; Larotonda, Gabriel Andrés; Recht, Lázaro; Varela, Alejandro; The left invariant metric in the general linear group; Elsevier Science; Journal Of Geometry And Physics; 86; 12-2014; 241-257
0393-0440
url http://hdl.handle.net/11336/12163
identifier_str_mv Andruchow, Esteban; Larotonda, Gabriel Andrés; Recht, Lázaro; Varela, Alejandro; The left invariant metric in the general linear group; Elsevier Science; Journal Of Geometry And Physics; 86; 12-2014; 241-257
0393-0440
dc.language.none.fl_str_mv eng
language eng
dc.relation.none.fl_str_mv info:eu-repo/semantics/altIdentifier/url/http://www.sciencedirect.com/science/article/pii/S0393044014001879
info:eu-repo/semantics/altIdentifier/doi/10.1016/j.geomphys.2014.08.009
info:eu-repo/semantics/altIdentifier/url/https://arxiv.org/abs/1109.0520
dc.rights.none.fl_str_mv info:eu-repo/semantics/openAccess
https://creativecommons.org/licenses/by-nc-nd/2.5/ar/
eu_rights_str_mv openAccess
rights_invalid_str_mv https://creativecommons.org/licenses/by-nc-nd/2.5/ar/
dc.format.none.fl_str_mv application/pdf
application/pdf
application/pdf
application/pdf
application/pdf
dc.publisher.none.fl_str_mv Elsevier Science
publisher.none.fl_str_mv Elsevier Science
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 13.070432

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