Before revision:  In this paper, we show how these computations in the most general case reduce to finding a canonical form of a matrix associated with a type-A quiver representation, which in turn can be computed using factorizations of associated matrices.
After revision:  In this paper, we show how these computations in the most general case reduce to finding a canonical form of a matrix associated with a type A quiver representation, which in turn can be computed using factorizations of associated matrices.