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Module « scipy.linalg »

Fonction subspace_angles - module scipy.linalg

Signature de la fonction subspace_angles

def subspace_angles(A, B) 

Description

help(scipy.linalg.subspace_angles)

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
-------
angles : ndarray, 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]``.

.. versionadded:: 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 :math:`\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

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