April 2025
Intermediate to advanced
328 pages
8h 42m
Portuguese (Portugal, Brazil)
Content preview from Álgebra linear prática para a ciência dos dados
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Capítulo 9. Matrizes ortogonais e decomposição QR
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Neste livro, aprenderás cinco decomposições principais: a decomposição ortogonal de vectores, a decomposição QR, a decomposição LU, a decomposição eigend e a decomposição do valor singular. Estas não são as únicas decomposições em álgebra linear, mas são as mais importantes para a ciência dos dados e a aprendizagem automática.
Neste capítulo, vais aprender QR. E, ao longo do caminho, aprenderá um novo tipo especial de matriz (ortogonal). A decomposição QR é um cavalo de batalha que potencializa aplicações, incluindo a inversa de matrizes, o ajuste de modelos de mínimos quadrados e a decomposição de eigend. Por isso, compreender e familiarizar-se com a decomposição QR vai ajudar-te a melhorar as tuas competências em álgebra linear.
Matrizes ortogonais
Começarei por te apresentar as matrizes ortogonais. Uma matriz ortogonal é uma matriz especial que é importante para várias decomposições, incluindo QR, eigendecomposição e decomposição do valor singular. A letra é frequentemente utilizada para indicar as matrizes ortogonais. As matrizes ortogonais têm duas propriedades:
- Colunas ortogonais
-
Todas as colunas são ortogonais aos pares.
- Colunas de norma unitária
-
A norma (comprimento geométrico) de cada coluna é exatamente 1.
Podemos traduzir estas duas propriedades numa expressão matemática (lembra-te que
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