Let A be an n × n matrix
defined over a field F of characteristic greater than n. For each n × n matrix X we
define
 (1) 
for each positive integer h. Then X is defined to be kcommutative with A if and
only if
 (2) 
Let P(x) be a polynomial such that P(A)≠0. Specifically, assume that
 (3) 
where p is a positive integer, each λ_{i} is a scalar from F, and λ_{p}≠0. In this paper we
study, for each positive integer k, the matrices X such that
 (4) 
We specify a polynomial P(A) in the form (3) and show how the maximal value of
k for which (4) has a solution depends on the polynomial P(A). In Theorem 3 it is
assumed that A is nonderogatory. Since the only matrices which commute with A in
this case are polynomials in A, we are, in effect, establishing a more precise bound for
k in (2) by predetermining X_{k}.
In the derogatory case, a matrix which is not a polynomial in A may commute
with A. However, Theorem 4 shows that if we choose a polynomial P(A) as X_{k}, then
the maximal value of k depends on the polynomial P.
