Reciprocal matrices: properties and approximation by a transitive matrix
Autor(a) principal: | |
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Data de Publicação: | 2020 |
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/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. |
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7160 |
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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 |
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Agência para a Sociedade do Conhecimento (UMIC) - FCT - Sociedade da Informação |
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RCAAP |
institution |
RCAAP |
reponame_str |
Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) |
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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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