September 2024
Intermediate to advanced
328 pages
8h 54m
Spanish
Content preview from Álgebra lineal práctica para la ciencia de datos
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Capítulo 9. Matrices ortogonales y descomposición QR
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En este libro aprenderás cinco descomposiciones principales: la descomposición ortogonal de vectores, la descomposición QR, la descomposición LU, la eigendecomposición y la descomposición del valor singular. No son las únicas descomposiciones del álgebra lineal, pero son las más importantes para la ciencia de datos y el aprendizaje automático.
En este capítulo aprenderás QR. Y por el camino, aprenderás un nuevo tipo especial de matriz (ortogonal). La descomposición QR es un caballo de batalla que potencia aplicaciones como la inversa de matrices, el ajuste de modelos por mínimos cuadrados y la eigendecomposición. Por tanto, comprender y familiarizarte con la descomposición QR te ayudará a mejorar tus conocimientos de álgebra lineal.
Matrices ortogonales
Empezaré presentándote las matrices ortogonales. Una matriz ortogonal es una matriz especial que es importante para varias descomposiciones, como la QR, la eigendecomposición y la descomposición en valores singulares. La letra se utiliza a menudo para indicar matrices ortogonales. Las matrices ortogonales tienen dos propiedades:
- Columnas ortogonales
-
Todas las columnas son ortogonales por pares.
- Columnas de norma unitaria
-
La norma (longitud geométrica) de cada columna es exactamente 1.
Podemos traducir esas dos propiedades en una expresión matemática ...
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