`numpy.linalg:svd`

source: /numpy/linalg/_linalg.py :1668

Signature

   def     svd (    a    ,    full_matrices    =  True   ,    compute_uv    =  True   ,    hermitian    =  False     )

Summary

Singular Value Decomposition.

Extended Summary

When a is a 2D array, and full_matrices=False, then it is factorized as u @ np.diag(s) @ vh = (u * s) @ vh, where u and the Hermitian transpose of vh are 2D arrays with orthonormal columns and s is a 1D array of a's singular values. When a is higher-dimensional, SVD is applied in stacked mode as explained below.

Parameters

a : (..., M, N) array_like: A real or complex array with a.ndim >= 2.
full_matrices : bool, optional: If True (default), u and vh have the shapes (..., M, M) and (..., N, N), respectively. Otherwise, the shapes are (..., M, K) and (..., K, N), respectively, where K = min(M, N).
compute_uv : bool, optional: Whether or not to compute u and vh in addition to s. True by default.
hermitian : bool, optional: If True, a is assumed to be Hermitian (symmetric if real-valued), enabling a more efficient method for finding singular values. Defaults to False.

Returns

U : { (..., M, M), (..., M, K) } array: Unitary array(s). The first a.ndim - 2 dimensions have the same size as those of the input a. The size of the last two dimensions depends on the value of full_matrices. Only returned when compute_uv is True.
S : (..., K) array: Vector(s) with the singular values, within each vector sorted in descending order. The first a.ndim - 2 dimensions have the same size as those of the input a.
Vh : { (..., N, N), (..., K, N) } array: Unitary array(s). The first a.ndim - 2 dimensions have the same size as those of the input a. The size of the last two dimensions depends on the value of full_matrices. Only returned when compute_uv is True.

Raises

: LinAlgError: If SVD computation does not converge.

Notes

When compute_uv is True, the result is a namedtuple with the following attribute names: U, S, and Vh.

The decomposition is performed using LAPACK routine _gesdd.

SVD is usually described for the factorization of a 2D matrix $A$ . The higher-dimensional case will be discussed below. In the 2D case, SVD is written as $A = U S V^{H}$ , where $A = a$ , $U = u$ , $S = np.diag (s)$ and $V^{H} = v h$ . The 1D array s contains the singular values of a and u and vh are unitary. The rows of vh are the eigenvectors of $A^{H} A$ and the columns of u are the eigenvectors of $A A^{H}$ . In both cases the corresponding (possibly non-zero) eigenvalues are given by s**2.

If a has more than two dimensions, then broadcasting rules apply, as explained in routines.linalg-broadcasting. This means that SVD is working in "stacked" mode: it iterates over all indices of the first a.ndim - 2 dimensions and for each combination SVD is applied to the last two indices. The matrix a can be reconstructed from the decomposition with either (u * s[..., None, :]) @ vh or u @ (s[..., None] * vh). (The @ operator can be replaced by the function np.matmul for python versions below 3.5.)

If a is a matrix object (as opposed to an ndarray), then so are all the return values.

Examples

import numpy as np
rng = np.random.default_rng()
a = rng.normal(size=(9, 6)) + 1j*rng.normal(size=(9, 6))
b = rng.normal(size=(2, 7, 8, 3)) + 1j*rng.normal(size=(2, 7, 8, 3))

✓

Reconstruction based on full SVD, 2D case:

U, S, Vh = np.linalg.svd(a, full_matrices=True)
U.shape, S.shape, Vh.shape
np.allclose(a, np.dot(U[:, :6] * S, Vh))
smat = np.zeros((9, 6), dtype=complex)
smat[:6, :6] = np.diag(S)
np.allclose(a, np.dot(U, np.dot(smat, Vh)))

✓

Reconstruction based on reduced SVD, 2D case:

U, S, Vh = np.linalg.svd(a, full_matrices=False)
U.shape, S.shape, Vh.shape
np.allclose(a, np.dot(U * S, Vh))
smat = np.diag(S)
np.allclose(a, np.dot(U, np.dot(smat, Vh)))

✓

Reconstruction based on full SVD, 4D case:

U, S, Vh = np.linalg.svd(b, full_matrices=True)
U.shape, S.shape, Vh.shape
np.allclose(b, np.matmul(U[..., :3] * S[..., None, :], Vh))
np.allclose(b, np.matmul(U[..., :3], S[..., None] * Vh))

✓

Reconstruction based on reduced SVD, 4D case:

U, S, Vh = np.linalg.svd(b, full_matrices=False)
U.shape, S.shape, Vh.shape
np.allclose(b, np.matmul(U * S[..., None, :], Vh))
np.allclose(b, np.matmul(U, S[..., None] * Vh))

✓

Aliases

numpy.linalg.svd