Quiz 2

Numerical Linear Algebra

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# Numerical Linear Algebra ## 7.1 QR Decomposition $\mathbf{X} = \mathbf{Q}\mathbf{R}$ where $\mathbf{Q}$ is orthogonal, $\mathbf{R}$ is upper triangular. Used for numerically stable least squares: $\hat{\boldsymbol{\beta}} = \mathbf{R}^{-1}\mathbf{Q}^T\mathbf{y}$ ## 7.2 Singular Value Decomposition $\mathbf{X} = \m...

Numerical Linear Algebra

7.1 QR Decomposition

X=QR\mathbf{X} = \mathbf{Q}\mathbf{R} where Q\mathbf{Q} is orthogonal, R\mathbf{R} is upper triangular. Used for numerically stable least squares: β^=R−1QTy\hat{\boldsymbol{\beta}} = \mathbf{R}^{-1}\mathbf{Q}^T\mathbf{y}
python
import numpy as np
X = np.random.randn(100, 5)
y = np.random.randn(100)
Q, R = np.linalg.qr(X)
beta = np.linalg.solve(R, Q.T @ y)

7.2 Singular Value Decomposition

X=UΣVT\mathbf{X} = \mathbf{U}\mathbf{\Sigma}\mathbf{V}^T — used for PCA, pseudoinverse, and handling rank deficiency.

7.3 Cholesky Decomposition

For positive definite A=LLT\mathbf{A} = \mathbf{L}\mathbf{L}^T (lower triangular). Used in Kalman filters and GP regression. Join Discord PreviousStochastic ProcessesNextOptimization Methods
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