# The Cayley-Hamilton Theorem

Terminology. A linear transformation T from a vector space V to itself (i.e. ) is called a linear operator on V.

Theorem. (Cayley-Hamilton) Let be a linear operator on a finite dimensional vector space V. Let p be the characteristic polynomial of T. Then .

Proof. Choose a basis for V. I will show that by showing that for all i.

Let . Then

Now

To save writing, let

Observe that the matrix has linear operators as its entries. For example, for ,

In fact, B is just the transpose of with . Hence, .

Next, I will show that for all k. Observe that

Hence,

This equation holds for all i and all k, so I'll still get 0 if I sum on i. So I'll sum on i, then interchange the order of summation:

Now is the -th entry of . Hence,

Since kills for all k, .

Definition. If A is an matrix, the minimal polynomial of A is the polynomial of smallest degree with leading coefficient 1 such that . If T is a linear operator on a vector space V, the minimal polynomial of T is the minimal polynomial of any matrix for T.

It's implicit in the last sentence that it doesn't matter which matrix for T you use. Can you prove it?

Corollary. The minimal polynomial divides the characteristic polynomial.

Example. Consider the matrix

The characteristic polynomial is ; the eigenvalues are (double) and .

Since A is evidently neither 0 nor a multiple of the identity, its minimal polynomial must be a quadratic or cubic factor of the characteristic polynomial.

Note that

Hence, the minimal polynomial is the characteristic polynomial .

Here is a more precise version of the previous corollary.

Proposition. Let be a linear operator on a finite dimensional vector space. The minimal and characteristic polynomials of T have the same roots, up to multiplicity.

Proof. Let denote the minimal polynomial and the characteristic polynomial. Cayley-Hamilton says that , so a root of m is a root of p.

Conversely, let be a root of p --- i.e. an eigenvalue. Let v be an eigenvector corresponding to , so . It follows that if is an arbitrary polynomial over F, then . In particular, this is true of the minimal polynomial:

Since , . Therefore, every root of p is a root of m, and the roots of m and p coincide.

Example. Consider the matrix

The characteristic polynomial is . In view of the Corollary, I did more work than necessary in determining the minimal polynomial the first time. The only possibilities for the minimal polynomial are and .

Computation showed that doesn't kill A, so the minimal polynomial is .