On the parametrization of the controllability subspaces of a controllable pair

Given a controllable linear control system defined by a pair of constant matrices (A, B), the set of controllability subspaces is a stratified submanifold of the set of (A, B)-invariant subspaces. We parametrize each strata by means of coordinate charts. This parametrization has significant differences to that of (A, B)-invariant subspaces, showing a more complex geometric structure.