5.5. THE CAYLEY HAMILTON THEOREM∗ 79

Note that each entry in C (λ ) is a polynomial in λ having degree no more than n− 1. Forexample, you might have something like

C (λ ) =

 λ2−6λ +9 3−λ 02λ −6 λ

2−3λ 0λ −1 λ −1 λ

2−3λ +2



=

 9 3 0−6 0 0−1 −1 2

+λ

 −6 −1 02 −3 01 1 −3

+λ2

 1 0 00 1 00 0 1

Therefore, collecting the terms in the general case,

C (λ ) =C0 +C1λ + · · ·+Cn−1λn−1

for C j some n×n matrix. Then

C (λ )(λ I−A) =(

C0 +C1λ + · · ·+Cn−1λn−1)(λ I−A) = q(λ ) I

Then multiplying out the middle term, it follows that for all |λ | sufficiently large,

a0I +a1Iλ + · · ·+ Iλn =C0λ +C1λ

2 + · · ·+Cn−1λn

−[C0A+C1Aλ + · · ·+Cn−1Aλ

n−1]

=−C0A+(C0−C1A)λ +(C1−C2A)λ2 + · · ·+(Cn−2−Cn−1A)λ

n−1 +Cn−1λn

Then, using Corollary 5.5.3, one can replace λ on both sides with A. Then the right side isseen to equal 0. Hence the left side, q(A) I is also equal to 0.

5.5.1 An Identity of Cauchy

Theorem 5.5.5 Both the left and the right sides in the following yield the same polynomialin the variables ai,bi for i≤ n.

∏i, j

(ai +b j)

∣∣∣∣∣∣∣1

a1+b1· · · 1

a1+bn...

...1

an+b1· · · 1

an+bn

∣∣∣∣∣∣∣= ∏j<i

(ai−a j)(bi−b j) . (5.5.16)

Proof: The theorem is true if n = 2. This follows from some computations. Suppose it