a(n) = trace(B(A^n))

[Edwin Clark]

In discussing Creighton Dement's floretion generated
sequences with him I came across the following fact:

Every sequence a(n) defined by a  k-th order linear recurrence 
with constant coefficients over any commutative ring with identity
can be realized in the form 

     a(n) = trace(B(A^n))

where A and B are k x k matrices over the ring in question.

Surely this must be known! Can someone provide a reference?

[Taking A to be the companion matrix of the polynomial defining
the recurrence will clearly give a sequence satisfying
a recurrence relation of order k with the desired coefficients
then one just needs to fiddle with B to get the desired initial
terms. One can take all entries of B to be 0 except for the
last column and it is not too hard to find a formula for the
last column of B in terms of the coefficients and initial 
conditions.]

--Edwin