scipy.linalg.svdvals

scipy.linalg.svdvals(a, overwrite_a=False, check_finite=True)

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

svdvals(a) only differs from svd(a, compute_uv=False) by its handling of the edge case of empty a, where it returns an empty sequence:

>>> a = np.empty((0, 2))
>>> from scipy.linalg import svdvals
>>> svdvals(a)
array([], dtype=float64)

Examples

>>> 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])

We can verify the maximum singular value of m by computing the maximum length of m.dot(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.dot(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
>>> np.random.seed(123)
>>> orth = ortho_group.rvs(4)
>>> svdvals(orth)
array([ 1.,  1.,  1.,  1.])