# 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'.

New in version 0.18.

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.

svdvals

Compute singular values of a matrix.

diagsvd

Construct the Sigma matrix, given the vector s.

Examples

>>> from scipy import linalg
>>> m, n = 9, 6
>>> a = np.random.randn(m, n) + 1.j*np.random.randn(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


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