Reachability matrices and cyclic matrices

Abstract

We study reachability matrices R(A, b) = [b, Ab,.. .,An−1b], where A is an n × n matrix over a field K and b is in Kn. We characterize those matrices that are reachability matrices for some pair (A, b). In the case of a cyclic matrix A and an n-vector of indeterminates x, we derive a factorization of the polynomial detR(A, x).