  Multipartite pure states are equivalent under Stochastic Local Operations and
Classical Communication (SLOCC) whenever they can be mapped into one another by
Invertible Local Operations. It is shown that this is equivalent to the
inter-convertibility through finite sequences of Elementary Local Operations. A
multipartite version of the Gauss-Jordan elimination strategy is then obtained,
enabling the analytical SLOCC-classification of previously unknown examples. It
is argued that the problem of SLOCC-classification is equivalent to the problem
of classifying the Multipartite Fully Reduced Forms that the coefficient matrix
of a state can assume after being subjected to this Gauss-Jordan elimination
procedure. The method is applied to examples of so-called LME and hypergraph
states, showing new SLOCC-equivalences. Moreover, possible physical
implications for states with a Dicke-like structure are sketched.
