# numpy.linalg.svd#

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, where K = 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.

New in version 1.17.0.

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.

scipy.linalg.svd

Similar function in SciPy.

scipy.linalg.svdvals

Compute singular values of a matrix.

Notes

Changed in version 1.8.0: Broadcasting rules apply, see the numpy.linalg documentation for details.

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 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

>>> a = np.random.randn(9, 6) + 1j*np.random.randn(9, 6)
>>> b = np.random.randn(2, 7, 8, 3) + 1j*np.random.randn(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