Given a pair of matrices (A,B), with every (A,B)-invariant subspace and the corresponding supplementary subspace we can associate pairs (A(1),B-1) and (A(2),B-2) which identify them up to feedback equivalence. A complete characterization of the controllability indices of (A,B) in terms of those of (A(1),B-1), (A(2),B-2) is provided under the assumption of complete controllability of (A(1),B-1). This problem is related to the feedback simulation problem.