numpy.linalg.
svd
Singular Value Decomposition.
When a is a 2D array, it is factorized as u @ np.diag(s) @ vh = (u * s) @ vh, where u and 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.
u @ np.diag(s) @ vh = (u * s) @ vh
A real or complex array with a.ndim >= 2.
a.ndim >= 2
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).
(..., M, M)
(..., N, N)
(..., M, K)
(..., K, N)
K = min(M, N)
Whether or not to compute u and 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.
New in version 1.17.0.
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.
a.ndim - 2
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.
If SVD computation does not converge.
Notes
Changed in version 1.8.0: Broadcasting rules apply, see the numpy.linalg documentation for details.
numpy.linalg
The decomposition is performed using LAPACK routine _gesdd.
_gesdd
SVD is usually described for the factorization of a 2D matrix
System Message: WARNING/2 (A)
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System Message: WARNING/2 (A = U S V^H)
System Message: WARNING/2 (A = a)
System Message: WARNING/2 (U= u)
System Message: WARNING/2 (S= \mathtt{np.diag}(s))
System Message: WARNING/2 (V^H = vh)
System Message: WARNING/2 (A^H A)
System Message: WARNING/2 (A A^H)
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.)
(u * s[..., None, :]) @ vh
u @ (s[..., None] * vh)
@
np.matmul
If a is a matrix object (as opposed to an ndarray), then so are all the return values.
matrix
ndarray
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