TITLE: Applications of the Cayley-Hamilton theorem
QUESTION [46 upvotes]: The Cayley-Hamilton theorem is usually presented in standard undergraduate courses in linear algebra as an important result. Recall that it says that any square matrix is a "root" of its own characteristic polynomial.
Question. Does this theorem have important applications?
Being not an algebraist, I am aware of only one application of this result which I would call important; it is very basic for commutative algebra, algebraic geometry, and number theory. It is as follows. Let a commutative unital ring $A$ be imbedded into a field $K$. Consider the set of elements of $K$ which are integral over $A$, i.e. are roots of a polynomial with coefficients in $A$ with the leading coefficient equal to 1. Then this set is a subring of $K$.

REPLY [4 votes]: There is a very important theorem by Procesi that derives all polynomial identities of the ring of matrices from the  Cayley-Hamilton theorem. For a review of rings with polynomial identities see this paper by Procesi.