April 2025
Intermediate to advanced
328 pages
8h 55m
Italian
Content preview from Algebra lineare pratica per la scienza dei dati
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Capitolo 9. Matrici ortogonali e decomposizione QR
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In questo libro imparerai cinque scomposizioni principali: scomposizione vettoriale ortogonale, scomposizione QR, scomposizione LU, scomposizione eigendecomposizione e scomposizione del valore singolare. Queste non sono le uniche scomposizioni dell'algebra lineare, ma sono le più importanti per la scienza dei dati e l'apprendimento automatico.
In questo capitolo imparerai a conoscere le QR. E lungo il percorso imparerai un nuovo tipo speciale di matrice (ortogonale). La decomposizione QR è un cavallo di battaglia che permette di realizzare applicazioni come l'inversione di matrici, l'adattamento di modelli ai minimi quadrati e l'eigendecomposizione. Pertanto, comprendere e acquisire familiarità con la decomposizione QR ti aiuterà a migliorare le tue competenze in algebra lineare.
Matrici ortogonali
Inizierò presentandoti le matrici ortogonali. Una matrice ortogonale è una matrice speciale che è importante per diverse decomposizioni, tra cui la QR, l'eigendecomposizione e la decomposizione del valore singolare. La lettera viene spesso utilizzata per indicare le matrici ortogonali. Le matrici ortogonali hanno due proprietà:
- Colonne ortogonali
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Tutte le colonne sono ortogonali a coppie.
- Colonne con norma unitaria
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La norma (lunghezza geometrica) di ciascuna colonna è esattamente 1.
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