numpy.linalg.svd
-
numpy.linalg.svd(a, full_matrices=True, compute_uv=True, hermitian=False)
[source] -
Singular Value Decomposition.
When
a
is a 2D array, it is factorized asu @ np.diag(s) @ vh = (u * s) @ vh
, whereu
andvh
are 2D unitary arrays ands
is a 1D array ofa
’s singular values. Whena
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
andvh
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
andvh
in addition tos
. 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 inputa
. The size of the last two dimensions depends on the value offull_matrices
. Only returned whencompute_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 inputa
. -
vh{ (…, N, N), (…, K, N) } array
-
Unitary array(s). The first
a.ndim - 2
dimensions have the same size as those of the inputa
. The size of the last two dimensions depends on the value offull_matrices
. Only returned whencompute_uv
is True.
-
- Raises
-
- LinAlgError
-
If SVD computation does not converge.
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
. The higher-dimensional case will be discussed below. In the 2D case, SVD is written as
, where
,
,
and
. The 1D array
s
contains the singular values ofa
andu
andvh
are unitary. The rows ofvh
are the eigenvectors ofand the columns of
u
are the eigenvectors of. 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 matrixa
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 amatrix
object (as opposed to anndarray
), 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
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