The Cayley-Hamilton theorem states that every square matrix over a commutative ring satisfies its own characteristic equation. That is, with the identity matrix, the characteristic polynomial of
is such that . I recalled that in a post a while ago, I mentioned that for any matrix , , a fact that is not hard to visualize based on calculation of determinants via minors, which is in fact much of what brings the existence of this adjugate to reason in some sense. This can be used to prove the Cayley-Hamilton theorem.
So we have
where is the characteristic polynomial of . The adjugate in the above is a matrix of polynomials in with coefficients that are matrices which are polynomials in , which we can represent in the form .
Equating coefficients gives us
With this, we have
with the RHS telescoping and annihilating itself to .
There is generalized version of this for a module over a ring, which goes as follows.
Cayley-Hamilton theorem (for modules) Let be a commutative ring with unity, a finitely generated -module, an ideal of , an endomorphism of with .
Proof: It’s mostly the same. Let be a generating set. Then for every , , with , with the s in . This means by closure properties of ideals the polynomial coefficients in the above will stay in . ▢
From this follows easily a statement of Nakayama’s lemma, ubiquitous in commutative algebra.
Nakayama’s lemma Let be an ideal in , and a finitely-generated module over . If , then there exists an with , such that .
Proof: With reference to the Cayley-Hamilton theorem, take , the identity map on , and define the polynomial as above. Then
both annihilates the s, coefficients residing in , so that and gives the zero map on in order for . ▢