| Name: | Description: | Size: | Format: | |
|---|---|---|---|---|
| 940.19 KB | Adobe PDF |
Advisor(s)
Abstract(s)
O conceito de fator invariante de uma matriz foi introduzido no séc. XIX por Smith. Segundo o seu teorema, qualquer matriz de números inteiros é equivalente a uma matriz diagonal, cujas entradas principais formam uma cadeia de divisibilidade, chamada forma normal de Smith. Foi
depois generalizado para domínios de ideais principais comutativos. Notou-se também que o conceito se aplica a classes de anéis mais gerais. Dados os fatores invariantes de duas matrizes, procurou-se determinar os possíveis fatores invariantes da sua soma e do seu produto. Introduzimos os resultados principais do tema, primeiro apenas em domínios de ideais principais comutativos, depois estendendo para todos os domínios de integridade comutativos onde uma forma normal de Smith existe sempre (denominados domínios de divisores elementares). Posteriormente, observamos que alguns resultados são válidos no caso não comutativo. No caso do produto, observamos o recurso à localização nos primos, assim como o papel das sequências de Littlewood-Richardson. Notamos a semelhança com outros problemas, nomeadamente, os valores próprios da soma de matrizes hermíticas, e os valores singulares da soma e do produto de matrizes. Mostramos como certas propriedades de divisibilidade em anéis comutativos também são válidas em anéis em que todos os ideais esquerdos e ideais direitos são bilaterais (anéis duo), tendo como consequência que alguns teoremas relativos a fatores invariantes são aplicáveis nestes anéis.
The concept of invariant factor of a matrix was introduced in the 19th century by Smith. According to his theorem, any matrix of integers is equivalent to a diagonal matrix, whose entries form a divisibility chain, called Smith normal form. It was then generalized to commutative Principal Ideal Domains. It was also noted the concept applies to more general classes of rings. Given the invariant factors of two matrices, attempts were made to determine the possible invariant factors of their sum and product. We introduce the main results of this topic, first referring only to commutative principal ideal domains, then extending to all commutative integral domains where a Smith normal form always exists (denominated elementary divisor domains). Later on, we note that some results are valid in the non-commutative case. In the case of the product, we observe the use of localization at primes, as well as the role of Littlewood-Richardson sequences. We note the similarity with other problems, namely, the eigenvalues of the sum of Hermitian matrices, and the singular values of the sum and the product of matrices. We show how certain properties of divisibility in commutative rings are also valid for rings where all left ideals and right ideals are bilateral (duo rings), and consequently, some theorems regarding invariant factors can be extended to such rings.
The concept of invariant factor of a matrix was introduced in the 19th century by Smith. According to his theorem, any matrix of integers is equivalent to a diagonal matrix, whose entries form a divisibility chain, called Smith normal form. It was then generalized to commutative Principal Ideal Domains. It was also noted the concept applies to more general classes of rings. Given the invariant factors of two matrices, attempts were made to determine the possible invariant factors of their sum and product. We introduce the main results of this topic, first referring only to commutative principal ideal domains, then extending to all commutative integral domains where a Smith normal form always exists (denominated elementary divisor domains). Later on, we note that some results are valid in the non-commutative case. In the case of the product, we observe the use of localization at primes, as well as the role of Littlewood-Richardson sequences. We note the similarity with other problems, namely, the eigenvalues of the sum of Hermitian matrices, and the singular values of the sum and the product of matrices. We show how certain properties of divisibility in commutative rings are also valid for rings where all left ideals and right ideals are bilateral (duo rings), and consequently, some theorems regarding invariant factors can be extended to such rings.
Description
Tese de Mestrado, Matemática, 2021, Universidade de Lisboa, Faculdade de Ciências
Keywords
Fatores invariantes Forma normal de Smith Anéis de divisores elementares Domínios de divisores elementares Anéis duo Teses de mestrado - 2021