scipy.linalg.svd

scipy.linalg.svd(a, full_matrices=True, compute_uv=True, overwrite_a=False)

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 : ndarray

Matrix to decompose, of shape (M,N).

full_matrices : bool, optional

If True, 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_uv : bool, optional

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

overwrite_a : bool, optional

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

Returns :

U : ndarray

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

s : ndarray

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

Vh : ndarray

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

>>> from scipy import linalg
>>> a = np.random.randn(9, 6) + 1.j*np.random.randn(9, 6)
>>> U, s, Vh = linalg.svd(a)
>>> U.shape, Vh.shape, s.shape
((9, 9), (6, 6), (6,))

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

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