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

A pair (A, B), where A is an n × n matrix and B is an n × 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[BAB … Aj−1 B] − rank[BAB … Aj−2B], 2 ≤ j ≤ p. Given a partial upper triangular matrix A of size n × n in upper canonical form and an n × m matrix B, we develop an algorithm that obtains a completion Ac of A, such that the pair (Ac, B) has an r-numbers sequence prescribed under some restrictions.