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