The Characteristic Polynomial
Given a linear endomorphism on a vector space of dimension , we’ve defined a function — — whose zeroes give exactly those field elements so that the -eigenspace of is nontrivial. Actually, we’ll switch it up a bit and use the function , which has the same useful property. Now let’s consider this function a little more deeply.
First off, if we choose a basis for we have matrix representations of endomorphisms, and thus a formula for their determinants. For instance, if is represented by the matrix with entries , then its determinant is given by
which is a sum of products of matrix entries. Now, the matrix entries for the transformation are given by . Each of these new entries is a polynomial (either constant or linear) in the variable . Any sum of products of polynomials is again a polynomial, and so our function is actually a polynomial in . We call it the “characteristic polynomial” of the transformation . In terms of the matrix entries of itself, we get
What’s the degree of this polynomial? Well, first let’s consider the degree of each term in the sum. Given a permutation the term is the product of factors. The th of these factors will be a field element if , and will be a linear polynomial if . Since multiplying polynomials adds their degrees, the degree of the term will be the number of such that . Thus the highest possible degree happens if for all index values . This only happens for one permutation — the identity — so there can’t be another term of the same degree to cancel the highest-degree monomial when we add them up. And so the characteristic polynomial has degree , equal to the dimension of the vector space .
What’s the leading coefficient? Again, the degree- monomial can only show up once, in the term corresponding to the identity permutation. Specifically, this term is
Each factor gives a coefficient of , and so the coefficient of the term is also . Thus the leading coefficient of the characteristic polynomial is — a fact which turns out to be useful.