numpy.linalg.svd — NumPy v2.2 Manual (original) (raw)

linalg.svd(a, full_matrices=True, compute_uv=True, hermitian=False)[source]#

Singular Value Decomposition.

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_matricesbool, 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, whereK = min(M, N).

compute_uvbool, optional

Whether or not to compute u and vh in addition to s. True by default.

hermitianbool, 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:

When compute_uv is True, the result is a namedtuple with the following

attribute names:

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

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 \(A = a\), \(U= u\),\(S= \mathtt{np.diag}(s)\) and \(V^H = vh\). 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 Linear algebra on several matrices at once. This means that SVD is working in “stacked” mode: it iterates over all indices of the firsta.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 oru @ (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)) + 1jrng.normal(size=(9, 6)) b = rng.normal(size=(2, 7, 8, 3)) + 1jrng.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 ((9, 9), (6,), (6, 6)) np.allclose(a, np.dot(U[:, :6] * S, Vh)) True smat = np.zeros((9, 6), dtype=complex) smat[:6, :6] = np.diag(S) np.allclose(a, np.dot(U, np.dot(smat, Vh))) True

Reconstruction based on reduced SVD, 2D case:

U, S, Vh = np.linalg.svd(a, full_matrices=False) U.shape, S.shape, Vh.shape ((9, 6), (6,), (6, 6)) np.allclose(a, np.dot(U * S, Vh)) True smat = np.diag(S) np.allclose(a, np.dot(U, np.dot(smat, Vh))) True

Reconstruction based on full SVD, 4D case:

U, S, Vh = np.linalg.svd(b, full_matrices=True) U.shape, S.shape, Vh.shape ((2, 7, 8, 8), (2, 7, 3), (2, 7, 3, 3)) np.allclose(b, np.matmul(U[..., :3] * S[..., None, :], Vh)) True np.allclose(b, np.matmul(U[..., :3], S[..., None] * Vh)) True

Reconstruction based on reduced SVD, 4D case:

U, S, Vh = np.linalg.svd(b, full_matrices=False) U.shape, S.shape, Vh.shape ((2, 7, 8, 3), (2, 7, 3), (2, 7, 3, 3)) np.allclose(b, np.matmul(U * S[..., None, :], Vh)) True np.allclose(b, np.matmul(U, S[..., None] * Vh)) True