Determinants and Their Applications in Mathematical Physics

80 4. Particular Determinants

ωris also a function ofn, but thenis suppressed to simplify the notation.

Thennumbers

1 ,ωr,ω

2

r,...,ω

n− 1

r (4.4.5)

are thenth roots of unity for any value ofr. Two different choices ofrgive

rise to the same set of roots but in a different order. It follows from the

third line in (4.4.4) that

n− 1

∑

s=0

ω

s

r

=0, 0 ≤r≤n− 1. (4.4.6)

Theorem.

An=

n− 1

∏

r=0

n

∑

s=1

ω

s− 1

r as.

Proof. Let

zr=

n

∑

s=1

ω

s− 1

r as

=a 1 +ωra 2 +ω

2

r

a 3 +···+ω

n− 1

r

an,ω

n

r

=1. (4.4.7)

Then,

ωrzr=an+ωra 1 +ω

2

ra^2 +···+ω

n− 1

r an−^1

ω

2

rzr=an−^1 +ωran+ω

2

ra^1 +···+ω

n− 1

r an−^2

............................................

ω

n− 1

r zr=a^2 +ωra^3 +ω

2

ra^4 +···+ω

n− 1

r a^1











. (4.4.8)

ExpressAnin column vector notation and perform a column operation:

An=

∣

∣C

1 C 2 C 3 ···Cn

∣

∣

=

∣

∣C′

1

C 2 C 3 ···Cn

∣

∣,

where

C

′

1 =

n

∑

j=1

ω

j− 1

r Cj

=

     

a 1

an

an− 1

.

.

.

a 2

     

+ωr

     

a 2

a 1

an

.

.

.

a 3

     

+ω

2

r

     

a 3

a 2

a 1

.

.

.

a 4

     

+···+ω

n− 1

r

     

an

an− 1

an− 2

.

.

.

a 1

     

=zrWr,

where

Wr=

[

1 ωrω

2

r

···ω

n− 1

r

]T

. (4.4.9)