Determinants and Their Applications in Mathematical Physics

4.4 Circulants 81

Hence,

A=zr

∣

∣

WrC 2 C 3 ···Cn

∣

∣

. (4.4.10)

It follows that eachzr,0≤r≤n−1, is a factor ofAn. Hence,

An=K

n− 1

∏

r=0

zr, (4.4.11)

but sinceAnand the product are homogeneous polynomials of degreen

in thear, the factorKmust be purely numerical. It is clear by comparing

the coefficients ofa

n

1 on each side thatK= 1. The theorem follows from

(4.4.7).

Illustration.Whenn=3,ωr= exp(2riπ/3),ω
3
r

=1.

ω 0 =1,

ω=ω 1 = exp(2iπ/3),

ω 2 = exp(4iπ/3) =ω

2

1 =ω

2

= ̄ω,

ω

2

2 =ω^1 =ω.

Hence,

A 3 =

∣ ∣ ∣ ∣ ∣ ∣

a 1 a 2 a 3

a 3 a 1 a 2

a 2 a 3 a 1

∣ ∣ ∣ ∣ ∣ ∣

=(a 1 +a 2 +a 3 )(a 1 +ω 1 a 2 +ω

2

1

a 3 )(a 1 +ω 2 a 2 +ω

2

2

a 3 )

=(a 1 +a 2 +a 3 )(a 1 +ωa 2 +ω

2

a 3 )(a 1 +ω

2

a 2 +ωa 3 ). (4.4.12)

Exercise.FactorizeA 4.

4.4.3 The Generalized Hyperbolic Functions

Define a matrixWas follows:

W=

[

ω

(r−1)(s−1)

]

n

(ω=ω 1 )

=

      

11 1 1··· 1

1 ωω

2

ω

3

··· ω

n− 1

1 ω

2

ω

4

ω

6

··· ω

2 n− 2

1 ω

3

ω

6

ω

9

··· ω

3 n− 3

··· ··· ··· ··· ··· ···

1 ω

n− 1

ω

2 n− 2

ω

3 n− 3

··· ω

(n−1)

2

       n

. (4.4.13)

Lemma 4.18.

W

− 1

=

1

n

W.

Proof.

W=

[

ω

−(r−1)(s−1)

]

n

.