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.

The documentation is written assuming array arguments are of specified “core” shapes. However, array argument(s) of this function may have additional “batch” dimensions prepended to the core shape. In this case, the array is treated as a batch of lower-dimensional slices; see Batched Linear Operations for details.

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