A.2.9 The Cayley Hamilton Theorem
Definition A.2.16 Let A be an n×n matrix. The characteristic polynomial is
and the solutions to qA
are called eigenvalues. For A a matrix and
+ a0, denote by p
the matrix defined by
The explanation for the last term is that A0 is interpreted as I, the identity
The Cayley Hamilton theorem states that every matrix satisfies its characteristic equation,
that equation defined by qA
= 0. It is one of the most important theorems in linear
The proof in this section is not the most general proof, but works well when the field of
scalars is ℝ
. The following lemma will help with its proof.
Lemma A.2.17 Suppose for all
where the Ai are n × n matrices. Then each Ai = 0.
Proof: Suppose some Ai≠0. Let p be the largest index of those which are non zero.
Then multiply by λ−p.
Now let λ →∞. Thus Ap = 0 after all. Hence each Ai = 0. ■
With the lemma, here is a simple corollary.
Corollary A.2.18 Let Ai and Bi be n × n matrices and suppose
large enough. Then Ai
= Bi for all i. If Ai
= Bi for each Ai,Bi then
one can substitute an n × n matrix M for λ and the identity will continue to
Proof: Subtract and use the result of the lemma. The last claim is obvious by
matching terms. ■
With this preparation, here is a relatively easy proof of the Cayley Hamilton
Theorem A.2.19 Let A be an n×n matrix and let q
be the characteristic polynomial. Then q
Proof: Let C
equal the transpose of the cofactor matrix of
is large enough, then
cannot be in the finite list of eigenvalues of A
and so for
exists.) Therefore, by Theorem A.2.14
Note that each entry in C
is a polynomial in
having degree no more than n−
example, you might have something like
Therefore, collecting the terms in the general case,
for Cj some n × n matrix. Then
Then multiplying out the middle term, it follows that for all
Then, using Corollary A.2.18, one can replace λ on both sides with A. Then
the right side is seen to equal 0. Hence the left side, q
is also equal to 0.