Extract a math-scale real-parameter vector corresponding to a special orthogonal matrix

Computes the principal matrix logarithm of a special orthogonal matrix `o_mat`, then returns the strictly lower-triangular entries of the resulting skew-symmetric matrix. This is a partial inverse of `build_orthogonal_matrix`, up to the periodicity of the exponential map.

Usage

extract_orthogonal_matrix_parameters(o_mat)

Arguments

o_mat

A \(k \times k\) special orthogonal matrix (t(o_mat) %*% o_mat = I and det(o_mat) = 1).

Value

A numeric vector of length \(k(k-1)/2\) containing the strictly lower-triangular entries of the skew-symmetric generator. For \(k=1\), returns NULL (the identity matrix).

Details

The matrix exponential \(\exp: \mathfrak{so}(k) \to SO(k)\) is surjective but not injective: different skew-symmetric matrices can exponentiate to the same orthogonal matrix. This function uses the **principal matrix logarithm** as implemented in `expm::logm`. Consequently, it may fail (or produce complex results) for matrices that have eigenvalues equal to \(-1\) (i.e., rotations by \(\pi\)). Such matrices lie on the cut locus of the exponential map and do not possess a unique real logarithm. If you encounter this, consider perturbing the matrix slightly away from the problematic rotation.

Examples

o_par2 <- 0.25
O2 <- build_orthogonal_matrix(o_par2)
extract_orthogonal_matrix_parameters(O2)
#> [1] 0.25