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 asu @ 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.
New in version 1.17.0.
- 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.
See also
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
oru @ (s[..., None] * vh)
. (The@
operator can be replaced by the functionnp.matmul
for python versions below 3.5.)If a is a
matrix
object (as opposed to anndarray
), 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 ((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