Page 95 - Determinants and Their Applications in Mathematical Physics

80 4. Particular Determinants

ω r is also a function of n, but the n is suppressed to simplify the notation.
The n numbers
2
1,ω r ,ω ,...,ω n−1 (4.4.5)
r r
are the nth roots of unity for any value of r. Two different choices of r give
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

ω =0, 0 ≤ r ≤ n − 1. (4.4.6)
s
r
s=0
Theorem.
n−1 n
s−1
A n = ω a s .
r
r=0 s=1
Proof. Let
n
s−1
z r = ω
r a s
s=1
2 n−1
= a 1 + ω r a 2 + ω a 3 + ··· + ω a n , ω =1. (4.4.7)
n
r r r
Then,
2
ω r z r = a n + ω r a 1 + ω a 2 + ··· + ω n−1 
r 2 r n−1
2
a n−1 
ω z r = a n−1 + ω r a n + ω a 1 + ··· + ω a n−2 
. (4.4.8)
r r r
............................................ 
2

ω n−1 z r = a 2 + ω r a 3 + ω a 4 + ··· + ω n−1 a 1
r r r
Express A n in column vector notation and perform a column operation:

A n = C 1 C 2 C 3 ··· C n

= C C 2 C 3 ··· C n ,
1

where
n
j−1

C = ω
1 r C j
j=1
       
a 1 a 2 a 3 a n
 a n   a 1   a 2   a n−1 
       
=  a n−1   a n  + ω 2  a 1  + ··· + ω n−1  a n−2 
. . . .
  + ω r   r   r  
. . . .
 .   .   .   . 
a 2 a 3 a 4 a 1
= z r W r ,
where
2
W r = 1 ω r ω ··· ω n−1 T . (4.4.9)
r r