Qualitative Controllability and Uncontrollability by a Single Entry

We characterize matrices A ∈ Cn×n whose zero/nonzero pattern requires that the controllability matrix [b Ab Ab . . . An−1b] ∈ Cn×n is of full rank, where b ∈ Cn×1 has exactly one nonzero entry. When all the diagonal entries of A are nonzero we show that this occurs if and only if QAQ is unreduced upper Hessenberg, with Q being a permutation matrix for which Qb = [b1, 0, . . . , 0] T . We also characterize matrices A whose zero/nonzero pattern requires that the controllability matrix is of deficient rank.