"""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).
"""
import geomstats.backend as gs
from geomstats.geometry.lie_algebra import MatrixLieAlgebra
from geomstats.geometry.matrices import Matrices
TOLERANCE = 1e-8
[docs]class SkewSymmetricMatrices(MatrixLieAlgebra):
"""Class for skew-symmetric matrices.
Parameters
----------
n : int
Number of rows and columns.
"""
def __init__(self, n):
dimension = int(n * (n - 1) / 2)
super(SkewSymmetricMatrices, self).__init__(dimension, n)
self.basis = gs.zeros((dimension, 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=TOLERANCE):
"""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: TOLERANCE.
Returns
-------
belongs : array-like, shape=[...,]
Boolean evaluating if matrix is skew symmetric.
"""
return Matrices(self.n, self.n).is_skew_symmetric(mat=mat, atol=atol)
[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.
"""
old_shape = gs.shape(matrix_representation)
as_vector = gs.reshape(matrix_representation, (old_shape[0], -1))
upper_tri_indices = gs.reshape(
gs.arange(0, self.n ** 2), (self.n, self.n)
)[gs.triu_indices(self.n, k=1)]
return as_vector[:, upper_tri_indices]