scipy.linalg.subspace_angles#

scipy.linalg.subspace_angles(A, B)[source]#

Compute the subspace angles between two matrices.

Parameters:

A(M, N) array_like: The first input array.
B(M, K) array_like: The second input array.

Returns:

anglesndarray, shape (min(N, K),): The subspace angles between the column spaces of A and B in descending order.

See also

orth
svd

Notes

This computes the subspace angles according to the formula provided in [1]. For equivalence with MATLAB and Octave behavior, use angles[0].

Added in version 1.0.

References

[1]

Knyazev A, Argentati M (2002) Principal Angles between Subspaces in an A-Based Scalar Product: Algorithms and Perturbation Estimates. SIAM J. Sci. Comput. 23:2008-2040.

Examples

An Hadamard matrix, which has orthogonal columns, so we expect that the suspace angle to be \(\frac{\pi}{2}\):

>>> import numpy as np
>>> from scipy.linalg import hadamard, subspace_angles
>>> rng = np.random.default_rng()
>>> H = hadamard(4)
>>> print(H)
[[ 1  1  1  1]
 [ 1 -1  1 -1]
 [ 1  1 -1 -1]
 [ 1 -1 -1  1]]
>>> np.rad2deg(subspace_angles(H[:, :2], H[:, 2:]))
array([ 90.,  90.])

And the subspace angle of a matrix to itself should be zero:

>>> subspace_angles(H[:, :2], H[:, :2]) <= 2 * np.finfo(float).eps
array([ True,  True], dtype=bool)

The angles between non-orthogonal subspaces are in between these extremes:

>>> x = rng.standard_normal((4, 3))
>>> np.rad2deg(subspace_angles(x[:, :2], x[:, [2]]))
array([ 55.832])  # random