svd(a, full_matrices=True, compute_uv=True, hermitian=False)
When a is a 2D array, it is factorized as u @ np.diag(s) @ vh
= (u * s) @ vh
, where u and :None:None:`vh` are 2D unitary arrays 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.
Broadcasting rules apply, see the :None:None:`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 :None:None:`vh` are unitary. The rows of :None:None:`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.
A real or complex array with a.ndim >= 2
.
If True (default), u and :None:None:`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)
.
Whether or not to compute u and :None:None:`vh` in addition to s. True by default.
If True, a is assumed to be Hermitian (symmetric if real-valued), enabling a more efficient method for finding singular values. Defaults to False.
If SVD computation does not converge.
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 :None:None:`full_matrices`. Only returned when :None:None:`compute_uv` is True.
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.
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 :None:None:`full_matrices`. Only returned when :None:None:`compute_uv` is True.
Singular Value Decomposition.
scipy.linalg.svd
Similar function in SciPy.
scipy.linalg.svdvals
Compute singular values of a matrix.
>>> 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)) TrueSee :
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