Module « scipy.linalg »

Fonction subspace_angles - module scipy.linalg

Signature de la fonction subspace_angles

def subspace_angles(A, B) 

Description

subspace_angles.__doc__

    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}`:

    >>> from numpy.random import default_rng
    >>> from scipy.linalg import hadamard, subspace_angles
    >>> rng = 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