Proof

From f | (X^d - 1) and f(A) = 0 we conclude A^d = I. The transition matrix for d shifts is A^d. Thus, after d shifts, we recover the initial situation.

If e is also a natural number such that after e shifts, we recover the initial situation, then so is gcd(d,e). (For, if A^e = A^d = I, then A^ae+bd = I for any two integers a, b. Of course, such a, b can be found using the extended Euclidean algorithm.) Therefore, the period, being the minimal such number, is a divisor of d.