Advanced book on Mathematics Olympiad

2.3 Linear Algebra 83

2.3.7 The Cayley–Hamilton and Perron–Frobenius Theorems........

We devote this section to two more advanced results, which seem to be relevant to
mathematics competitions. All matrices below are assumed to have complex entries.

The Cayley–Hamilton Theorem.Anyn×nmatrixAsatisfies its characteristic equa-
tion, which means that ifPA(λ)=det(λIn−A),thenPA(A)=On.

Proof.LetPA(λ)=λn+an− 1 λn−^1 + ··· +a 0. Denote by(λIn−A)the adjoint of
(λIn−A)(the one used in the computation of the inverse). Then

(λIn−A)(λIn−A)∗=det(λIn−A)In.

The entries of the adjoint matrix(λIn−A)∗are polynomials inλof degree at mostn−1.
Splitting the matrix by the powers ofλ, we can write

(λIn−A)∗=Bn− 1 λn−^1 +Bn− 2 λn−^2 +···+B 0.

Equating the coefficients ofλon both sides of

(λIn−A)(Bn− 1 λn−^1 +Bn− 2 λn−^2 +···+B 0 )=det(λIn−A)In,

we obtain the equations

Bn− 1 =In,

−ABn− 1 +Bn− 2 =an− 1 In,

−ABn− 2 +Bn− 3 =an− 2 In,

···

−AB 0 =a 0 In.

Multiply the first equation byAn, the second byAn−^1 , the third byAn−^2 , and so on, then
add then+1 equations to obtain

On=An+an− 1 An−^1 +an− 2 An−^2 +···+a 0 In.

This equality is just the desiredPA(A)=On. 

As a corollary we prove the trace identity for SL( 2 ,C)matrices. This identity is
important in the study of characters of group representations.

Example.LetAandBbe 2×2 matrices with determinant equal to 1. Prove that

tr(AB)−(trA)(trB)+tr(AB−^1 )= 0.