There are two main forms of representation for elements of a
matrix Lie algebra implemented here. The first one is as a matrix, as
elements of \(R^{n \times n}\). The second is by choosing a basis and
remembering the coefficients of an element in that basis. This basis will
be provided in child classes (e.g. SkewSymmetricMatrices).
Super classes
rgeomstats::PythonClass -> rgeomstats::Manifold -> rgeomstats::VectorSpace -> MatrixLieAlgebra
Public fields
nAn integer value representing the number of rows and columns in the matrix representation of the Lie algebra.
Methods
Inherited methods
rgeomstats::PythonClass$get_python_class()rgeomstats::PythonClass$set_python_class()rgeomstats::Manifold$belongs()rgeomstats::Manifold$is_tangent()rgeomstats::Manifold$random_point()rgeomstats::Manifold$random_tangent_vec()rgeomstats::Manifold$regularize()rgeomstats::Manifold$set_metric()rgeomstats::Manifold$to_tangent()rgeomstats::VectorSpace$projection()
Method new()
The MatrixLieAlgebra class constructor.
Usage
MatrixLieAlgebra$new(dim, n, ..., py_cls = NULL)Arguments
dimAn integer value specifying the dimension of the Lie algebra as a real vector space.
nAn integer value representing the number of rows and columns in the matrix representation of the Lie algebra.
...Extra arguments to be passed to parent class constructors. See
VectorSpaceandManifoldclasses.py_clsA Python object of class
MatrixLieAlgebra. Defaults toNULLin which case it is instantiated on the fly using the other input arguments.
Method baker_campbell_hausdorff()
Calculates the Baker-Campbell-Hausdorff approximation of given order.
Arguments
matrix_aA numeric array of shape \(... \times n \times n\) specifying a matrix or a sample of matrices.
matrix_bA numeric array of shape \(... \times n \times n\) specifying a matrix or a sample of matrices.
orderAn integer value specifying the order to which the approximation is calculated. Note that this is NOT the same as using only \(e_i\) with \(i < \mathrm{order}\). Defaults to
2L.
Details
The implementation is based on casas2009efficient;textualrgeomstats with the pre-computed constants taken from casas2009data;textualrgeomstats. Our coefficients are truncated to enable us to calculate BCH up to order \(15\). This represents $$Z = \log \left( \exp(X) \exp(Y) \right)$$ as an infinite linear combination of the form $$Z = \sum_i z_i e_i$$ where \(z_i\) are rational numbers and \(e_i\) are iterated Lie brackets starting with \(e_1 = X\), \(e_2 = Y\), each \(e_i\) is given by some \((i^\prime,i^{\prime\prime})\) such that \(e_i = [e_i^\prime, e_i^{\prime\prime}]\).
Method basis_representation()
Computes the coefficients of matrices in the given basis.
Method matrix_representation()
Compute the matrix representation for the given basis coefficients.