O Teorema de Perron-Frobenius
This work aimed to study the spectrum of a certain class of matrices. More specifically, we present a detailed proof of the Perron-Frobenius theorem in the context of this class of matrices, following the method developed by Weilandt. This is a research that has as its methodology the bibliograph...
| Autor principal: | Souza, Isis Costa de Paula e. |
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| Grau: | Monografia |
| Idioma: | pt_BR |
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Universidade Federal do Tocantins
2023
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| Acesso em linha: |
http://hdl.handle.net/11612/4731 |
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ir-11612-4731 |
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ir-11612-47312023-02-09T06:01:15Z O Teorema de Perron-Frobenius Souza, Isis Costa de Paula e. Mesquita, Élis Gardel da Costa Teorema de Perron-Frobenius Matriz Autovalor Autovetor CNPQ::CIENCIAS EXATAS E DA TERRA::MATEMATICA This work aimed to study the spectrum of a certain class of matrices. More specifically, we present a detailed proof of the Perron-Frobenius theorem in the context of this class of matrices, following the method developed by Weilandt. This is a research that has as its methodology the bibliographic review. Perron’s theorem was stated and proved in 1907, which guarantees the existence of a maximal eigenvalue and a strictly positive associated eigenvector, for the class of square matrices with positive entries. In 1912, Frobenius extended this result to the class of non-negative and irreducible matrices. The classical form of the theorem is presented in three which state the existence of a maximal eigenvalue r, the existence of an eigenvector v with all positive entries associated with r and the influence of the variation of A on the variation of r. In addition to the developed theory presenting interesting results, this theorem is very important in several applications in areas such as economics, demography, physics, probability, among others. Este trabalho teve como objetivo estudar o espectro de uma determinada classe de matrizes. Mais especificamente, apresentamos uma demonstração detalhada do teorema de Perron- Frobenius no contexto desta classe de matrizes, seguindo o método desenvolvido por Weilandt. Esta é uma pesquisa que tem como metodologia a revisão bibliográfica. O teorema de Perron foi enunciado e demonstrado em 1907, o qual garante a existência de autovalor maximal e autovetor associado estritamente positivo, para a classe das matrizes quadradas com entradas positivas. Em 1912, Frobenius ampliou este resultado à classe das matrizes não negativas e irredutíveis. A forma clássica do teorema é apresentada em três partes, as quais afirmam a existência de um valor próprio maximal r, a existência de um vetor próprio v com todas entradas positivas associado a r e a influência da variação das entradas de A sobre a variação de r. Além da teoria desenvolvida apresentar resultados interessantes, este teorema se faz muito importante em diversas aplicações em áreas tais como economia, demografia, física, probabilidade, entre outras. 2023-02-08T19:58:24Z 2023-02-08T19:58:24Z 2022-06-28 Monografia Souza, Isis Costa de Paula e. O Teorema de Perron-Frobenius. 42 f. Monografia de Graduação - Curso de Licenciatura em Matemática. Universidade Federal do Tocantins, Arraias, 2022. http://hdl.handle.net/11612/4731 pt_BR Acesso livre application/pdf Universidade Federal do Tocantins Arraias CURSO::ARRAIAS::PRESENCIAL::LICENCIATURA::MATEMÁTICA Arraias Graduação |
| institution |
Repositório Institucional - Universidade Federal do Tocantins - UFT |
| collection |
RepositorioUFT |
| language |
pt_BR |
| topic |
Teorema de Perron-Frobenius Matriz Autovalor Autovetor CNPQ::CIENCIAS EXATAS E DA TERRA::MATEMATICA |
| spellingShingle |
Teorema de Perron-Frobenius Matriz Autovalor Autovetor CNPQ::CIENCIAS EXATAS E DA TERRA::MATEMATICA Souza, Isis Costa de Paula e. O Teorema de Perron-Frobenius |
| topic_facet |
Teorema de Perron-Frobenius Matriz Autovalor Autovetor CNPQ::CIENCIAS EXATAS E DA TERRA::MATEMATICA |
| description |
This work aimed to study the spectrum of a certain class of matrices. More specifically,
we present a detailed proof of the Perron-Frobenius theorem in the context of this class of
matrices, following the method developed by Weilandt. This is a research that has as its
methodology the bibliographic review. Perron’s theorem was stated and proved in 1907,
which guarantees the existence of a maximal eigenvalue and a strictly positive associated
eigenvector, for the class of square matrices with positive entries. In 1912, Frobenius
extended this result to the class of non-negative and irreducible matrices. The classical
form of the theorem is presented in three which state the existence of a maximal eigenvalue
r, the existence of an eigenvector v with all positive entries associated with r and the
influence of the variation of A on the variation of r. In addition to the developed theory
presenting interesting results, this theorem is very important in several applications in
areas such as economics, demography, physics, probability, among others. |
| author_additional |
Mesquita, Élis Gardel da Costa |
| author_additionalStr |
Mesquita, Élis Gardel da Costa |
| format |
Monografia |
| author |
Souza, Isis Costa de Paula e. |
| title |
O Teorema de Perron-Frobenius |
| title_short |
O Teorema de Perron-Frobenius |
| title_full |
O Teorema de Perron-Frobenius |
| title_fullStr |
O Teorema de Perron-Frobenius |
| title_full_unstemmed |
O Teorema de Perron-Frobenius |
| title_sort |
o teorema de perron-frobenius |
| publisher |
Universidade Federal do Tocantins |
| publishDate |
2023 |
| url |
http://hdl.handle.net/11612/4731 |
| _version_ |
1787134373351391232 |
| score |
11.753735 |