- Suppose A is a linear transformation and let the characteristic polynomial be
where the ϕ

_{j}are irreducible. Explain using Corollary 7.3.11 why the irreducible factors of the minimal polynomial are ϕ_{j}and why the minimal polynomial is of the form ∏_{j=1}^{q}ϕ_{j}^{rj}where r_{j}≤ n_{j}. You can use the Cayley Hamilton theorem if you like. - Find the minimal polynomial for
by the above technique assuming the field of scalars is the rational numbers. Is what you found also the characteristic polynomial?

- Show, using the rational root theorem, the minimal polynomial for A in the above problem is irreducible with respect to ℚ. Letting the field of scalars be ℚ find the rational canonical form and a similarity transformation which will produce it.
- Letting the field of scalars be ℚ, find the rational canonical form for the matrix
- Let A : ℚ
^{3}→ ℚ^{3}be linear. Suppose the minimal polynomial is. Find the rational canonical form. Can you give generalizations of this rather simple problem to other situations? - Find the rational canonical form with respect to the field of scalars equal to ℚ for the
matrix
Observe that this particular matrix is already a companion matrix of λ

^{3}−λ^{2}+ λ− 1. Then find the rational canonical form if the field of scalars equals ℂ or ℚ + iℚ. - Let qbe a polynomial and C its companion matrix. Show the characteristic and minimal polynomial of C are the same and both equal q.
- ↑Use the existence of the rational canonical form to give a proof of the Cayley Hamilton theorem valid for any field, even fields like the integers mod p for p a prime. The earlier proof 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.
- Suppose you have two n × n matrices A,B whose entries are in a field F and suppose G is an extension of F. For example, you could have F = ℚ and G = ℂ. Suppose A and B are similar with respect to the field G. Can it be concluded that they are similar with respect to the field F? Hint: First show that the two have the same minimal polynomial over F. Next consider the proof of Lemma 9.8.3 and show that they have the same rational canonical form with respect to F.

Download PDFView PDF