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