scipy.linalg.

svdvals#

scipy.linalg.svdvals(a, overwrite_a=False, check_finite=True)[source]#

Compute singular values of a matrix.

Parameters:

a(M, N) array_like: Matrix to decompose.
overwrite_abool, optional: Whether to overwrite a; may improve performance. Default is False.
check_finitebool, optional: Whether to check that the input matrix contains only finite numbers. Disabling may give a performance gain, but may result in problems (crashes, non-termination) if the inputs do contain infinities or NaNs.

Returns:

s(min(M, N),) ndarray: The singular values, sorted in decreasing order.

Raises:

LinAlgError: If SVD computation does not converge.

See also

svd: Compute the full singular value decomposition of a matrix.
diagsvd: Construct the Sigma matrix, given the vector s.

Notes

Array argument of this function, a, may have additional “batch” dimensions prepended to the core shape. In this case, the array is treated as a batch of lower-dimensional slices; see Batched Linear Operations for details.

Examples

>>> import numpy as np
>>> from scipy.linalg import svdvals
>>> m = np.array([[1.0, 0.0],
...               [2.0, 3.0],
...               [1.0, 1.0],
...               [0.0, 2.0],
...               [1.0, 0.0]])
>>> svdvals(m)
array([ 4.28091555,  1.63516424])

If the input matrix has more than two dimensions, it is interpreted as a batch of two-dimensional matrices:

>>> mm = np.stack((m, 2*m))
>>> svdvals(mm)
array([[4.28091555, 1.63516424],
       [8.56183109, 3.27032847]])

We can verify the maximum singular value of m by computing the maximum length of m @ u over all the unit vectors u in the (x,y) plane. We approximate “all” the unit vectors with a large sample. Because of linearity, we only need the unit vectors with angles in [0, pi].

>>> t = np.linspace(0, np.pi, 2000)
>>> u = np.array([np.cos(t), np.sin(t)])
>>> np.linalg.norm(m @ u, axis=0).max()
4.2809152422538475

p is a projection matrix with rank 1. With exact arithmetic, its singular values would be [1, 0, 0, 0].

>>> v = np.array([0.1, 0.3, 0.9, 0.3])
>>> p = np.outer(v, v)
>>> svdvals(p)
array([  1.00000000e+00,   2.02021698e-17,   1.56692500e-17,
         8.15115104e-34])

The singular values of an orthogonal matrix are all 1. Here, we create a random orthogonal matrix by using the rvs() method of scipy.stats.ortho_group.

>>> from scipy.stats import ortho_group
>>> orth = ortho_group.rvs(4)
>>> svdvals(orth)
array([ 1.,  1.,  1.,  1.])