Definition

The characteristic polynomial of a linear map for a finite dimensional vector space over a field is

Note that is a monic polynomial of degree equal to the dimension of whose roots are the eigenvalues of .

Relation to rational canonical form

Let be an endomorphism of a finite dimensional vector space over with rational canonical form

Then .

We don’t have to do the whole matrix, it suffices to prove that . Write as

so that

We compute by induction on degree of . The base case () is

so we are good there.

In general, we expand along the first row to get