Hermite indices and equivalence relations

The Hermite indices are invariant for the right equivalence of non-singular polynomial matrices and for the similarity of controllable matrix pairs. Nevertheless, they do not form a complete system of invariants. The aim of this work is to define two equivalence relations, one in the set of non-singular polynomial matrices and the other one in the set of controllable matrix pairs for which the Hermite indices form a complete system of invariants.