# scipy.linalg.subspace_angles

scipy.linalg.subspace_angles(A, B)

  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]`.

  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 \(\frac{\pi}{2}\):

      >>> 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