An interlacing theorem for matrices whose graph is a given tree
Autor(a) principal: | |
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Data de Publicação: | 2006 |
Tipo de documento: | Artigo |
Idioma: | eng |
Título da fonte: | Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) |
Texto Completo: | http://hdl.handle.net/10316/7709 https://doi.org/10.1007/s10958-006-0394-1 |
Resumo: | Abstract Let A and B be (nn)-matrices. For an index set S ? {1, …, n}, denote by A(S) the principal submatrix that lies in the rows and columns indexed by S. Denote by S' the complement of S and define ?(A, B) = $$\mathop \sum \limits_S $$ det A(S) det B(S'), where the summation is over all subsets of {1, …, n} and, by convention, det A(Ø) = det B(Ø) = 1. C. R. Johnson conjectured that if A and B are Hermitian and A is positive semidefinite, then the polynomial ?(?A,-B) has only real roots. G. Rublein and R. B. Bapat proved that this is true for n ? 3. Bapat also proved this result for any n with the condition that both A and B are tridiagonal. In this paper, we generalize some little-known results concerning the characteristic polynomials and adjacency matrices of trees to matrices whose graph is a given tree and prove the conjecture for any n under the additional assumption that both A and B are matrices whose graph is a tree. |
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An interlacing theorem for matrices whose graph is a given treeAbstract Let A and B be (nn)-matrices. For an index set S ? {1, …, n}, denote by A(S) the principal submatrix that lies in the rows and columns indexed by S. Denote by S' the complement of S and define ?(A, B) = $$\mathop \sum \limits_S $$ det A(S) det B(S'), where the summation is over all subsets of {1, …, n} and, by convention, det A(Ø) = det B(Ø) = 1. C. R. Johnson conjectured that if A and B are Hermitian and A is positive semidefinite, then the polynomial ?(?A,-B) has only real roots. G. Rublein and R. B. Bapat proved that this is true for n ? 3. Bapat also proved this result for any n with the condition that both A and B are tridiagonal. In this paper, we generalize some little-known results concerning the characteristic polynomials and adjacency matrices of trees to matrices whose graph is a given tree and prove the conjecture for any n under the additional assumption that both A and B are matrices whose graph is a tree.2006info:eu-repo/semantics/publishedVersioninfo:eu-repo/semantics/articlehttp://hdl.handle.net/10316/7709http://hdl.handle.net/10316/7709https://doi.org/10.1007/s10958-006-0394-1engJournal of Mathematical Sciences. 139:4 (2006) 6823-6830Fonseca, C. M. dainfo: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:RCAAP2020-05-25T13:05:11Zoai:estudogeral.uc.pt:10316/7709Portal AgregadorONGhttps://www.rcaap.pt/oai/openaireopendoar:71602024-03-19T21:00:39.881183Repositório 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 |
An interlacing theorem for matrices whose graph is a given tree |
title |
An interlacing theorem for matrices whose graph is a given tree |
spellingShingle |
An interlacing theorem for matrices whose graph is a given tree Fonseca, C. M. da |
title_short |
An interlacing theorem for matrices whose graph is a given tree |
title_full |
An interlacing theorem for matrices whose graph is a given tree |
title_fullStr |
An interlacing theorem for matrices whose graph is a given tree |
title_full_unstemmed |
An interlacing theorem for matrices whose graph is a given tree |
title_sort |
An interlacing theorem for matrices whose graph is a given tree |
author |
Fonseca, C. M. da |
author_facet |
Fonseca, C. M. da |
author_role |
author |
dc.contributor.author.fl_str_mv |
Fonseca, C. M. da |
description |
Abstract Let A and B be (nn)-matrices. For an index set S ? {1, …, n}, denote by A(S) the principal submatrix that lies in the rows and columns indexed by S. Denote by S' the complement of S and define ?(A, B) = $$\mathop \sum \limits_S $$ det A(S) det B(S'), where the summation is over all subsets of {1, …, n} and, by convention, det A(Ø) = det B(Ø) = 1. C. R. Johnson conjectured that if A and B are Hermitian and A is positive semidefinite, then the polynomial ?(?A,-B) has only real roots. G. Rublein and R. B. Bapat proved that this is true for n ? 3. Bapat also proved this result for any n with the condition that both A and B are tridiagonal. In this paper, we generalize some little-known results concerning the characteristic polynomials and adjacency matrices of trees to matrices whose graph is a given tree and prove the conjecture for any n under the additional assumption that both A and B are matrices whose graph is a tree. |
publishDate |
2006 |
dc.date.none.fl_str_mv |
2006 |
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/10316/7709 http://hdl.handle.net/10316/7709 https://doi.org/10.1007/s10958-006-0394-1 |
url |
http://hdl.handle.net/10316/7709 https://doi.org/10.1007/s10958-006-0394-1 |
dc.language.iso.fl_str_mv |
eng |
language |
eng |
dc.relation.none.fl_str_mv |
Journal of Mathematical Sciences. 139:4 (2006) 6823-6830 |
dc.rights.driver.fl_str_mv |
info:eu-repo/semantics/openAccess |
eu_rights_str_mv |
openAccess |
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 |
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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 |
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1799133897092497408 |