On the parametrization of the controllability pairs of a controllable subspaces of a controllable pair
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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.