r-Numbers completion problems of partial upper canonical form II Original Research Article

A pair (A, B), where A is an n x n matrix and B is an n x m matrix, is said to have the nonnegative integers sequence {rj}j=1p as the r-numbers sequence if r1 = rank(B) and rj = rank[B AB ⋯ Aj−1B] − rank[B AB ⋯ Aj−2B], 2 ≤ j ≤ p. Given a partial upper triangular matrix A of size n x n in upper canonical form and an n x m matrix B, we develop algorithms that obtain a completion Ac of A, such that the pair (Ac, B) has an r-numbers sequence prescribed under some restrictions.