* 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 .

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Copyright 2011 by Bruce Ikenaga