svd — SciPy v1.15.2 Manual (original) (raw)

scipy.linalg.

scipy.linalg.svd(a, full_matrices=True, compute_uv=True, overwrite_a=False, check_finite=True, lapack_driver='gesdd')[source]#

Singular Value Decomposition.

Factorizes the matrix a into two unitary matrices U and Vh, and a 1-D array s of singular values (real, non-negative) such thata == U @ S @ Vh, where S is a suitably shaped matrix of zeros with main diagonal s.

Parameters:

a(M, N) array_like

Matrix to decompose.

full_matricesbool, optional

If True (default), U and Vh are of shape (M, M), (N, N). If False, the shapes are (M, K) and (K, N), whereK = min(M, N).

compute_uvbool, optional

Whether to compute also U and Vh in addition to s. Default is True.

overwrite_abool, optional

Whether to overwrite a; may improve performance. Default is False.

check_finitebool, optional

Whether to check that the input matrix contains only finite numbers. Disabling may give a performance gain, but may result in problems (crashes, non-termination) if the inputs do contain infinities or NaNs.

lapack_driver{‘gesdd’, ‘gesvd’}, optional

Whether to use the more efficient divide-and-conquer approach ('gesdd') or general rectangular approach ('gesvd') to compute the SVD. MATLAB and Octave use the 'gesvd' approach. Default is 'gesdd'.

Returns:

Undarray

Unitary matrix having left singular vectors as columns. Of shape (M, M) or (M, K), depending on full_matrices.

sndarray

The singular values, sorted in non-increasing order. Of shape (K,), with K = min(M, N).

Vhndarray

Unitary matrix having right singular vectors as rows. Of shape (N, N) or (K, N) depending on full_matrices.

For compute_uv=False, only s is returned.

Raises:

LinAlgError

If SVD computation does not converge.

See also

svdvals

Compute singular values of a matrix.

diagsvd

Construct the Sigma matrix, given the vector s.

Examples

import numpy as np from scipy import linalg rng = np.random.default_rng() m, n = 9, 6 a = rng.standard_normal((m, n)) + 1.j*rng.standard_normal((m, n)) U, s, Vh = linalg.svd(a) U.shape, s.shape, Vh.shape ((9, 9), (6,), (6, 6))

Reconstruct the original matrix from the decomposition:

sigma = np.zeros((m, n)) for i in range(min(m, n)): ... sigma[i, i] = s[i] a1 = np.dot(U, np.dot(sigma, Vh)) np.allclose(a, a1) True

Alternatively, use full_matrices=False (notice that the shape ofU is then (m, n) instead of (m, m)):

U, s, Vh = linalg.svd(a, full_matrices=False) U.shape, s.shape, Vh.shape ((9, 6), (6,), (6, 6)) S = np.diag(s) np.allclose(a, np.dot(U, np.dot(S, Vh))) True

s2 = linalg.svd(a, compute_uv=False) np.allclose(s, s2) True