I needed the following well-known result in the course I’m teaching:
Lemma. Let be a field and
a nilpotent matrix. Then
.
The classical proof uses the flag of subspaces
to produce a basis for which is upper triangular. Here is a slick basis-independent commutative algebra proof that shows something better:
Lemma. Let be a commutative ring with nilradical
, and let
be a nilpotent matrix. Then the characteristic polynomial
satisfies
Here we write for the polynomials in
of degree smaller than
whose coefficients lie in a given ideal
.
Note that the formulation should ring a bell: in the previous post we saw that . When
is a domain, this reduces to
, and the lemma just says that
.
This suggests that we shouldn’t work with but with its anadrome (or reciprocal)
.
Proof of Lemma. We have to show that . Since
is nilpotent, there exists
with
, so
. Thus
. Evaluating at
shows that the constant coefficient is 1.