scipy.linalg.

svd#

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 that a == 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), where K = 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 of U 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