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_a : bool, optional

Whether to overwrite a; may improve performance. Default is False.

check_finite : bool, 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.])