Definition A.2.16Let A be an n×n matrix. The characteristic polynomialisdefined as

qA (t) ≡ det(tI − A )

and the solutions to q_{A}

(t)

= 0 are called eigenvalues. For A a matrix andp

(t)

= t^{n} + a_{n−1}t^{n−1} +

⋅⋅⋅

+ a_{1}t + a_{0}, denote by p

(A)

the matrix defined by

p (A ) ≡ An + a An−1 + ⋅⋅⋅+ a A + a I.
n−1 1 0

The explanation for the last term is that A^{0}is interpreted as I, the identitymatrix.

The Cayley Hamilton theorem states that every matrix satisfies its characteristic equation,
that equation defined by q_{A}

(t)

= 0. It is one of the most important theorems in linear
algebra^{1} .
The proof in this section is not the most general proof, but works well when the field of
scalars is ℝ or ℂ. The following lemma will help with its proof.

Lemma A.2.17Suppose for all

|λ|

large enough,

m
A0 + A1λ+ ⋅⋅⋅+ Am λ = 0,

where the A_{i}are n × n matrices. Then each A_{i} = 0.

Proof:Suppose some A_{i}≠0. Let p be the largest index of those which are non zero.
Then multiply by λ^{−p}.

−p −p+1 −1
A0 λ + A1λ + ⋅⋅⋅+Ap −1λ + Ap = 0

Now let λ →∞. Thus A_{p} = 0 after all. Hence each A_{i} = 0. ■

With the lemma, here is a simple corollary.

Corollary A.2.18Let A_{i}and B_{i}be n × n matrices and suppose

m m
A0 + A1 λ+ ⋅⋅⋅+ Amλ = B0 + B1λ + ⋅⋅⋅+Bm λ

for all

|λ|

large enough. Then A_{i} = B_{i}for all i. If A_{i} = B_{i}for each A_{i},B_{i}thenone can substitute an n × n matrix M for λ and the identity will continue tohold.

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.

Theorem A.2.19Let A be an n×n matrix and let q

(λ)

≡ det

(λI − A)

be the characteristic polynomial. Then q

(A)

= 0.

Proof:Let C

(λ)

equal the transpose of the cofactor matrix of

(λI − A)

for

|λ|

large.
(If

|λ|

is large enough, then λ cannot be in the finite list of eigenvalues of A and so for
such λ,