"""Module providing the SkewSymmetricMatrices class.
This is the Lie algebra of the Special Orthogonal Group.
As basis we choose the matrices with a single 1 on the upper triangular part
of the matrices (and a -1 in its lower triangular part), except in dim 2 and
3 to match usual conventions.
"""
import geomstats.backend as gs
from geomstats.geometry.lie_algebra import MatrixLieAlgebra
[docs]class SkewSymmetricMatrices(MatrixLieAlgebra):
"""Class for skew-symmetric matrices.
Parameters
----------
n : int
Number of rows and columns.
"""
def __init__(self, n):
dim = int(n * (n - 1) / 2)
super(SkewSymmetricMatrices, self).__init__(dim, n)
if n == 2:
self.basis = gs.array([[[0., -1.], [1., 0.]]])
elif n == 3:
self.basis = gs.array([
[[0., 0., 0.],
[0., 0., -1.],
[0., 1., 0.]],
[[0., 0., 1.],
[0., 0., 0.],
[-1., 0., 0.]],
[[0., -1., 0.],
[1., 0., 0.],
[0., 0., 0.]]])
else:
self.basis = gs.zeros((dim, n, n))
basis = []
for row in gs.arange(n - 1):
for col in gs.arange(row + 1, n):
basis.append(gs.array_from_sparse(
[(row, col), (col, row)], [1., -1.], (n, n)))
self.basis = gs.stack(basis)
[docs] def belongs(self, mat, atol=gs.atol):
"""Evaluate if mat is a skew-symmetric matrix.
Parameters
----------
mat : array-like, shape=[..., n, n]
Square matrix to check.
atol : float
Tolerance for the equality evaluation.
Optional, default: backend atol.
Returns
-------
belongs : array-like, shape=[...,]
Boolean evaluating if matrix is skew symmetric.
"""
is_skew = self.is_skew_symmetric(mat=mat, atol=atol)
return gs.logical_and(
is_skew, super(SkewSymmetricMatrices, self).belongs(mat))
[docs] def random_point(self, n_samples=1, bound=1.):
"""Sample from a uniform distribution.
Parameters
----------
n_samples : int
Number of samples.
Optional, default: 1.
bound : float
Bound of the interval in which to sample each entry.
Optional, default: 1.
Returns
-------
point : array-like, shape=[m, n] or [n_samples, m, n]
Sample.
"""
return self.projection(self.random_point(n_samples, bound))
[docs] @classmethod
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 cls.to_skew_symmetric(mat)
[docs] def basis_representation(self, matrix_representation):
"""Calculate the coefficients of given matrix in the basis.
Compute a 1d-array that corresponds to the input matrix in the basis
representation.
Parameters
----------
matrix_representation : array-like, shape=[..., n, n]
Matrix.
Returns
-------
basis_representation : array-like, shape=[..., dim]
Representation in the basis.
"""
if self.n == 2:
return matrix_representation[..., 1, 0][..., None]
if self.n == 3:
vec = gs.stack([
matrix_representation[..., 2, 1],
matrix_representation[..., 0, 2],
matrix_representation[..., 1, 0]])
return gs.transpose(vec)
return gs.triu_to_vec(matrix_representation, k=1)