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lect17, Tue 03/10
Singular value decomposition
References for this week’s lectures
NCM Sections 10.1, 10.2, 10.10, 10.11
Outline
- Matrix factorizations, old and new:
- Gaussian elimination with partial pivoting: P @ A = L @ U
- Cholesky factorization of SPD matrix: A = R.T @ R
- Orthogonal factorization: A = Q @ R
- Eigenvalues: A @ X = X @ S
- Eigenvalues of SPD matrix: A = V @ S @ V.T
- Singular value decomposition: A = U @ S @ V.T
- Singular value decomposition:
- Geometry: m-by-n matrix A maps the unit sphere in Rn to an ellipsoid in Rm
- Two orthogonal bases, one in domain space Rn and one in range space Rm
- rank(A) = # of nonzero singular values
- range(A) and nullspace(A) from U and V
- 2-norm(A) = max singular value
- Frobenius norm of A = sqrt(sum(singular values))
- Determinant of A = product(singular values)
- A = sum_i (s_i * outerproduct(u_i, v_i))
- numpy/scipy routines:
- spla.svd()