linalg.svd

Description

Decompose a singular value.

When a is a 2D array, and full_matrices=False, 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, singular value decomposition (SVD) is applied in stacked mode.

Mandatory Input Parameters

Parameter	Type	Description
a	(…,M,N) array_like	A real or complex array with a.ndim >= 2
b	array_like	-

Optional Input Parameters

Parameter	Type	Default Value	Description
full_matrices	bool	True	If the default value True is used, 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_uv	bool	True	Whether or not to compute u and vh in addition to s.

Return Value

Parameter	Type	Description
u	{(…,M,M), (…,M,K)} array	Unitary array. 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. It is only returned when compute_uv is True.
s	(…, K) array	Vectors with 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. 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. It is only returned when compute_uv is True.

Examples

>>> import numpy as np
>>> 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)
>>> 
>>> 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
>>>

Parent topic: Linear Algebra Functions