def _iter_log(p, matrix_v, max_iter, tol):
matrix_lv = gs.zeros_like(matrix_v)
for _ in range(max_iter):
matrix_lv = gs.linalg.logm(matrix_v)
matrix_c = matrix_lv[..., p:, p:]
norm_matrix_c = gs.linalg.norm(matrix_c)
if norm_matrix_c <= tol:
break
matrix_phi = gs.linalg.expm(-Matrices.to_skew_symmetric(matrix_c))
aux_matrix = gs.matmul(matrix_v[..., :, p:], matrix_phi)
matrix_v = gs.concatenate([matrix_v[..., :, :p], aux_matrix],
axis=-1)
return matrix_lv
def projection(cls, mat):
r"""Compute the skew-symmetric component of a matrix.
The skew-symmetric part of a matrix :math: `X` is defined by
.. math:
(X - X^T) / 2
Parameters
----------
mat : array-like, shape=[..., n, n]
Matrix.
Returns
-------
skew_sym : array-like, shape=[..., n, n]
Skew-symmetric matrix.
"""
return Matrices.to_skew_symmetric(mat)
def to_tangent(self, vector, base_point=None):
"""Project a vector to a tangent space of the manifold.
Compute the bracket (commutator) of the base_point with
the skew-symmetric part of vector.
Parameters
----------
vector : array-like, shape=[..., n, n]
Vector.
base_point : array-like, shape=[..., n, n]
Point on the manifold.
Returns
-------
tangent_vec : array-like, shape=[..., n, n]
Tangent vector at base point.
"""
skew = Matrices.to_skew_symmetric(vector)
return Matrices.bracket(base_point, skew)
