Page 71 - Determinants and Their Applications in Mathematical Physics

56 4. Particular Determinants

This completes the proof of Theorem 4.3 which can be expressed in the
form

n
(y s − x i )
i=1
= .
H rs

n
(y s − x r ) (x r − x i )
X n
i=1
i =r
Let
(m)
A n = |σ |
i,j−1 n
(m) (m) (m)
σ σ ... σ
10 11
1,n−1
(m) (m) (m)

= σ 20 σ 21 ... σ 2,n−1 , m ≥ n.

........................
(m) (m) (m)
σ σ ... σ

n0 n1 n,n−1 n
Theorem 4.4.
A n =(−1) n(n−1)/2 X n .
Proof.

A n = C 0 C 1 C 2 ... C n−1 ,

n
where, from the lemma in Appendix A.7,

(m) (m) (m) (m) T
C j = σ σ σ ... σ
1j 2j 3j nj
j
(m) (m)
= σ v j−p v j−p v j−p ... v j−p T , v r = −x r ,σ =1.
p 1 2 3 n 0
p=0
Applying the column operations
j

(m)
C = C j − σ
j C j−k
k
k=1
in the order j =1, 2, 3,... so that each new column created by one operation
is applied in the next operation, it is found that

C = v v v ... v j T , j =0, 1, 2,... .
j
j
j
j 1 2 3 n
Hence
j−1
A n = |v
i | n
=(−1) n(n−1)/2 |x j−1 | n .
i
Theorem 4.4 follows.