Let be nonscalar matrices over a field . Prove that and are similar if and only if they have the same characteristic polynomial.

We showed in this previous exercise that similar matrices have the same characteristic polynomial. Thus it suffices to show that two nonscalar matrices of dimension 2 having the same characteristic polynomial are similar. To do this, it suffices to show that two such matrices have the same rational canonical form.

To this end, let and be matrices over having the same characteristic polynomial. Note that has degree 2.

Let be the minimal polynomial of . If has degree 1, then by Proposition 20 in D&F, and the invariant factors of are and . But now is similar to the diagonal matrix . But if , then in fact , a contradiction. So has degree 2. Since divides the characteristic polynomial of (being the divisibility-largest invariant factor, while the characteristic polynomial is the product of the invariant factors) and both are monic, we have .

Similarly, we have .

So and have the same minimal polynomial , which is in fact their (only) invariant factor. So and have the same rational canonical form. By Theorem 15 in D&F, and are similar.

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