Hermite indices and the action of the feedback group

The controllability indices are a complete system of invariants for the feedback equivalence relation of controllable matrix pairs. The Hermite indices are invariant for the similarity of matrix pairs but they are not invariant by changing bases on the input space and performing state feedback. The aim of this work is to partially characterize the Hermite indices of a controllable matrix pair when transformations of the feedback group are performed on it.