262 CHAPTER 9. CANONICAL FORMS

7. Let q (λ) be a polynomial and C its companion matrix. Show the characteristic andminimal polynomial of C are the same and both equal q (λ).

8. ↑Use the existence of the rational canonical form to give a proof of the Cayley Hamiltontheorem valid for any field, even fields like the integers mod p for p a prime. The earlierproof based on determinants was fine for fields like Q or R where you could let λ→ ∞but it is not clear the same result holds in general.

9. Suppose you have two n×n matrices A,B whose entries are in a field F and suppose Gis an extension of F. For example, you could have F = Q and G = C. Suppose A andB are similar with respect to the field G. Can it be concluded that they are similarwith respect to the field F? Hint: First show that the two have the same minimalpolynomial over F. Next consider the proof of Lemma 9.8.3 and show that they havethe same rational canonical form with respect to F.