My task is to find some manifold on stoichiometric matrices and find the geodesic distance between these nodes (not the euclidean distance).
Here's my idea so far: Suppose we are given a matrix N of size m by n and we multiply by a diagonal weighted flux matrix W of size n by n. We can perform SVD and get USVT. I want to take advantage of the geodesic on SO(n). Since U is orthogonal, its determinant is +1 or -1. If its +1, we are done. Otherwise, we do an row swap (by multiplying Permutation matrix), which flips the sign to +1. Does this whole idea make sense or are there any improvements to them? Thanks!