Reciprocal matrices: properties and approximation by a transitive matrix

Detalhes bibliográficos
Autor(a) principal: Bebiano, Natália
Data de Publicação: 2020
Outros Autores: Fernandes, Rosário, Furtado, Susana
Tipo de documento: Artigo
Idioma: eng
Título da fonte: Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos)
Texto Completo: http://hdl.handle.net/10362/116008
Resumo: N. Bebiano: partially supported by project UID/MAT/00324/2019. R. Fernandes: partially supported by project UID/MAT/00297/2019. S. Furtado: partially supported by project UID/MAT/04721/2019.
id RCAP_1d1e94e6a05649f7858fb86576617337
oai_identifier_str oai:run.unl.pt:10362/116008
network_acronym_str RCAP
network_name_str Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos)
repository_id_str 7160
spelling Reciprocal matrices: properties and approximation by a transitive matrixAnalytical hierarchical processFrobenius normPerron eigenvalueRankReciprocal matrixToeplitz matrixTransitive matrixComputational MathematicsApplied MathematicsN. Bebiano: partially supported by project UID/MAT/00324/2019. R. Fernandes: partially supported by project UID/MAT/00297/2019. S. Furtado: partially supported by project UID/MAT/04721/2019.Reciprocal matrices and, in particular, transitive matrices, appear in several applied areas. Among other applications, they have an important role in decision theory in the context of the analytical hierarchical process, introduced by Saaty. In this paper, we study the possible ranks of a reciprocal matrix and give a procedure to construct a reciprocal matrix with the rank and the off-diagonal entries of an arbitrary row (column) prescribed. We apply some techniques from graph theory to the study of transitive matrices, namely to determine the maximum number of equal entries, and distinct from ± 1 , in a transitive matrix. We then focus on the n-by-n reciprocal matrix, denoted by C(n, x), with all entries above the main diagonal equal to x> 0. We show that there is a Toeplitz transitive matrix and a transitive matrix preserving the maximum possible number of entries of C(n, x), whose distances to C(n, x), measured in the Frobenius norm, are smaller than the one of the transitive matrix proposed by Saaty, constructed from the right Perron eigenvector of C(n, x). We illustrate our results with some numerical examples.CMA - Centro de Matemática e AplicaçõesRUNBebiano, NatáliaFernandes, RosárioFurtado, Susana2022-02-10T01:30:39Z2020-05-012020-05-01T00:00:00Zinfo:eu-repo/semantics/publishedVersioninfo:eu-repo/semantics/articleapplication/pdfhttp://hdl.handle.net/10362/116008eng2238-3603PURE: 28462136https://doi.org/10.1007/s40314-020-1075-2info: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:RCAAP2024-03-11T04:58:33Zoai:run.unl.pt:10362/116008Portal AgregadorONGhttps://www.rcaap.pt/oai/openaireopendoar:71602024-03-20T03:42:54.565318Repositório 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 Reciprocal matrices: properties and approximation by a transitive matrix
title Reciprocal matrices: properties and approximation by a transitive matrix
spellingShingle Reciprocal matrices: properties and approximation by a transitive matrix
Bebiano, Natália
Analytical hierarchical process
Frobenius norm
Perron eigenvalue
Rank
Reciprocal matrix
Toeplitz matrix
Transitive matrix
Computational Mathematics
Applied Mathematics
title_short Reciprocal matrices: properties and approximation by a transitive matrix
title_full Reciprocal matrices: properties and approximation by a transitive matrix
title_fullStr Reciprocal matrices: properties and approximation by a transitive matrix
title_full_unstemmed Reciprocal matrices: properties and approximation by a transitive matrix
title_sort Reciprocal matrices: properties and approximation by a transitive matrix
author Bebiano, Natália
author_facet Bebiano, Natália
Fernandes, Rosário
Furtado, Susana
author_role author
author2 Fernandes, Rosário
Furtado, Susana
author2_role author
author
dc.contributor.none.fl_str_mv CMA - Centro de Matemática e Aplicações
RUN
dc.contributor.author.fl_str_mv Bebiano, Natália
Fernandes, Rosário
Furtado, Susana
dc.subject.por.fl_str_mv Analytical hierarchical process
Frobenius norm
Perron eigenvalue
Rank
Reciprocal matrix
Toeplitz matrix
Transitive matrix
Computational Mathematics
Applied Mathematics
topic Analytical hierarchical process
Frobenius norm
Perron eigenvalue
Rank
Reciprocal matrix
Toeplitz matrix
Transitive matrix
Computational Mathematics
Applied Mathematics
description N. Bebiano: partially supported by project UID/MAT/00324/2019. R. Fernandes: partially supported by project UID/MAT/00297/2019. S. Furtado: partially supported by project UID/MAT/04721/2019.
publishDate 2020
dc.date.none.fl_str_mv 2020-05-01
2020-05-01T00:00:00Z
2022-02-10T01:30:39Z
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/10362/116008
url http://hdl.handle.net/10362/116008
dc.language.iso.fl_str_mv eng
language eng
dc.relation.none.fl_str_mv 2238-3603
PURE: 28462136
https://doi.org/10.1007/s40314-020-1075-2
dc.rights.driver.fl_str_mv info:eu-repo/semantics/openAccess
eu_rights_str_mv openAccess
dc.format.none.fl_str_mv application/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
_version_ 1799138039861084160

Registros relacionados