# 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.

Notes

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

New 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 $$\frac{\pi}{2}$$:

>>> from numpy.random import default_rng
>>> from scipy.linalg import hadamard, subspace_angles
>>> rng = default_rng()
>>> print(H)
[[ 1  1  1  1]
[ 1 -1  1 -1]
[ 1  1 -1 -1]
[ 1 -1 -1  1]]
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))