The structure of matrices with a maximum multiplicity eigenvalue
Autor(a) principal: | |
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Data de Publicação: | 2008 |
Outros Autores: | , |
Tipo de documento: | Artigo |
Idioma: | eng |
Título da fonte: | Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) |
Texto Completo: | http://hdl.handle.net/10316/4579 https://doi.org/10.1016/j.laa.2008.04.016 |
Resumo: | There is remarkable and distinctive structure among Hermitian matrices, whose graph is a given tree T and that have an eigenvalue of multiplicity that is a maximum for T. Among such structure, we give several new results: (1) no vertex of T may be "neutral"; (2) neutral vertices may occur if the largest multiplicity is less than the maximum; (3) every Parter vertex has at least two downer branches; (4) removal of a Parter vertex changes the status of no other vertex; and (5) every set of Parter vertices forms a Parter set. Statements (3), (4) and (5) are also not generally true when the multiplicity is less than the maximum. Some of our results are used to give further insights into prior results, and both the review of necessary background and the development of new structural lemmas may be of independent interest. |
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The structure of matrices with a maximum multiplicity eigenvalueHermitian matricesEigenvaluesMultiplicitiesMaximum multiplicityPath cover numberParter verticesThere is remarkable and distinctive structure among Hermitian matrices, whose graph is a given tree T and that have an eigenvalue of multiplicity that is a maximum for T. Among such structure, we give several new results: (1) no vertex of T may be "neutral"; (2) neutral vertices may occur if the largest multiplicity is less than the maximum; (3) every Parter vertex has at least two downer branches; (4) removal of a Parter vertex changes the status of no other vertex; and (5) every set of Parter vertices forms a Parter set. Statements (3), (4) and (5) are also not generally true when the multiplicity is less than the maximum. Some of our results are used to give further insights into prior results, and both the review of necessary background and the development of new structural lemmas may be of independent interest.http://www.sciencedirect.com/science/article/B6V0R-4SMF2K9-2/1/e80ee40f33898e52f517c5d58dbfa5bc2008info:eu-repo/semantics/publishedVersioninfo:eu-repo/semantics/articleaplication/PDFhttp://hdl.handle.net/10316/4579http://hdl.handle.net/10316/4579https://doi.org/10.1016/j.laa.2008.04.016engLinear Algebra and its Applications. 429:4 (2008) 875-886Johnson, Charles R.Duarte, António LealSaiago, Carlos M.info:eu-repo/semantics/openAccessreponame:Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos)instname:Agência para a Sociedade do Conhecimento (UMIC) - FCT - Sociedade da Informaçãoinstacron:RCAAP2020-11-06T16:49:13Zoai:estudogeral.uc.pt:10316/4579Portal AgregadorONGhttps://www.rcaap.pt/oai/openaireopendoar:71602024-03-19T21:00:47.193961Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) - Agência para a Sociedade do Conhecimento (UMIC) - FCT - Sociedade da Informaçãofalse |
dc.title.none.fl_str_mv |
The structure of matrices with a maximum multiplicity eigenvalue |
title |
The structure of matrices with a maximum multiplicity eigenvalue |
spellingShingle |
The structure of matrices with a maximum multiplicity eigenvalue Johnson, Charles R. Hermitian matrices Eigenvalues Multiplicities Maximum multiplicity Path cover number Parter vertices |
title_short |
The structure of matrices with a maximum multiplicity eigenvalue |
title_full |
The structure of matrices with a maximum multiplicity eigenvalue |
title_fullStr |
The structure of matrices with a maximum multiplicity eigenvalue |
title_full_unstemmed |
The structure of matrices with a maximum multiplicity eigenvalue |
title_sort |
The structure of matrices with a maximum multiplicity eigenvalue |
author |
Johnson, Charles R. |
author_facet |
Johnson, Charles R. Duarte, António Leal Saiago, Carlos M. |
author_role |
author |
author2 |
Duarte, António Leal Saiago, Carlos M. |
author2_role |
author author |
dc.contributor.author.fl_str_mv |
Johnson, Charles R. Duarte, António Leal Saiago, Carlos M. |
dc.subject.por.fl_str_mv |
Hermitian matrices Eigenvalues Multiplicities Maximum multiplicity Path cover number Parter vertices |
topic |
Hermitian matrices Eigenvalues Multiplicities Maximum multiplicity Path cover number Parter vertices |
description |
There is remarkable and distinctive structure among Hermitian matrices, whose graph is a given tree T and that have an eigenvalue of multiplicity that is a maximum for T. Among such structure, we give several new results: (1) no vertex of T may be "neutral"; (2) neutral vertices may occur if the largest multiplicity is less than the maximum; (3) every Parter vertex has at least two downer branches; (4) removal of a Parter vertex changes the status of no other vertex; and (5) every set of Parter vertices forms a Parter set. Statements (3), (4) and (5) are also not generally true when the multiplicity is less than the maximum. Some of our results are used to give further insights into prior results, and both the review of necessary background and the development of new structural lemmas may be of independent interest. |
publishDate |
2008 |
dc.date.none.fl_str_mv |
2008 |
dc.type.status.fl_str_mv |
info:eu-repo/semantics/publishedVersion |
dc.type.driver.fl_str_mv |
info:eu-repo/semantics/article |
format |
article |
status_str |
publishedVersion |
dc.identifier.uri.fl_str_mv |
http://hdl.handle.net/10316/4579 http://hdl.handle.net/10316/4579 https://doi.org/10.1016/j.laa.2008.04.016 |
url |
http://hdl.handle.net/10316/4579 https://doi.org/10.1016/j.laa.2008.04.016 |
dc.language.iso.fl_str_mv |
eng |
language |
eng |
dc.relation.none.fl_str_mv |
Linear Algebra and its Applications. 429:4 (2008) 875-886 |
dc.rights.driver.fl_str_mv |
info:eu-repo/semantics/openAccess |
eu_rights_str_mv |
openAccess |
dc.format.none.fl_str_mv |
aplication/PDF |
dc.source.none.fl_str_mv |
reponame:Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) instname:Agência para a Sociedade do Conhecimento (UMIC) - FCT - Sociedade da Informação instacron:RCAAP |
instname_str |
Agência para a Sociedade do Conhecimento (UMIC) - FCT - Sociedade da Informação |
instacron_str |
RCAAP |
institution |
RCAAP |
reponame_str |
Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) |
collection |
Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) |
repository.name.fl_str_mv |
Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) - Agência para a Sociedade do Conhecimento (UMIC) - FCT - Sociedade da Informação |
repository.mail.fl_str_mv |
|
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1799133898399023104 |