Definition 21.4.1Let A be an n × n matrix. The characteristic polynomialis defined as
qA (t) ≡ det (tI − A )
and the solutions to q_{A}
(t)
= 0 are called eigenvalues. For A a matrix and p
(t)
= t^{n} + a_{n−1}t^{n−1} +
⋅⋅⋅
+ a_{1}t + a_{0},denote by p
(A)
the matrix defined by
n n−1
p (A ) ≡ A + an−1A +⋅⋅⋅+ a1A + a0I.
The explanation for the last term is that A^{0}is interpreted as I, the identity matrix.
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 21.4.2Suppose for all
|λ|
large enough,
m
A0 +A1 λ+ ⋅⋅⋅+ Am λ = 0,
where the A_{i}are n × n matrices. Then each A_{i} = 0.
Proof:Multiply by λ^{−m} to obtain
A λ−m + A λ−m+1 + ⋅⋅⋅+ A λ−1 + A = 0.
0 1 m−1 m
Now let
|λ|
→∞ to obtain A_{m} = 0. With this, multiply by λ to obtain
A0λ− m+1 + A1 λ−m+2 + ⋅⋅⋅+ Am −1 = 0.
Now let
|λ|
→∞ to obtain A_{m−1} = 0. Continue multiplying by λ and letting λ →∞ to obtain that all the
A_{i} = 0. ■
With the lemma, here is a simple corollary.
Corollary 21.4.3Let A_{i}and B_{i}be n × n matrices and suppose
m m
A0 + A1λ + ⋅⋅⋅+ Am λ = 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}then one can substitute ann × n matrix M for λ and the identity will continue to hold.
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 21.4.4Let A be an n × n matrix and let q
(λ)
≡ det
(λI − A)
be the characteristicpolynomial. 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,
Then a short computation shows that for all complex λ,
(λI + E )(λI + E ) = (λ2 + λ)I = (λI + E )(λI + E )
1 2 2 1
However,
(N I + E1)(NI + E2) ⁄= (N I + E2)(N I + E1)
The reason this can take place is that N fails to commute with E_{i}. Of course a scalar commutes with any
matrix so there was no difficulty in obtaining that the matrix equation held for arbitrary λ, but this
factored equation does not continue to hold if λ is replaced by a matrix. In the above proof of the Cayley
Hamilton theorem, this issue was avoided by considering only polynomials which are of the form
C_{0} + C_{1}λ +
⋅⋅⋅
in which the polynomial identity held because the corresponding matrix coefficients
were equal. However, you can also argue that in the above proof, the C_{i} each commute with
A.
Theorem 21.4.5Let q
(λ)
be the characteristic polynomial and p
(λ)
the minimal polynomial. Thenthere is a polynomial l
(λ)
which could be a constant such that q
(λ)
= l
(λ)
p
(λ)
.
Proof:By the division algorithm, q
(λ)
= p
(λ)
l
(λ)
+ r
(λ)
where the degree of r
(λ)
is less than the
degree of p
(λ)
or else r
(λ)
= 0. But then, substituting in A, you get r
(A)
= 0 which is impossible if its
degree is less than that of p