Characteristic polynomials and controlability of partially prescribed matrices

In a previous paper it was proved that n−1 arbitrary entries and the characteristic polynomial of a n×n matrix over a field F can be arbitrarily prescribed, except if all the nonprincipal entries of a row or column are prescribed equal to zero and the characteristic polynomial does not have a root in F. This paper describes the possible characteristic polynomials of a pk×pk matrix, partitioned into k×k blocks of size p×p when k−1 blocks are fixed and the others vary. It also studies the possibility of a pair of matrices (A1,A2), where A1 is square and is partitioned into k×(k+1) blocks of size p×p, being completely controllable when some of the blocks are prescribed and the others vary.