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() >>> 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