Page 287 - Determinants and Their Applications in Mathematical Physics

272 6. Applications of Determinants in Mathematical Physics
Proof.

1 j −1/2
D (φ i )= b e [(−1) λ i e i + µ i ]
j
j
x 2 i i
−1/2
so that every element in row i of W contains the factor e . Removing
i
all these factors from the determinant,
(e 1 e 2 ...e n ) 1/2 W
1 1
b 2
λ 1 e 1 + µ 1 2
2 1 (−λ 1 e 1 + µ 1 )( b 1 ) (λ 1 e 1 + µ 1 ) ···
1 1
b 2
λ 2 e 2 + µ 2 2
2 2 (−λ 2 e 2 + µ 2 )( b 2 ) (λ 2 e 2 + µ 2 ) ··· (6.7.45)
1 1 2
=
λ 3 e 3 + µ 3 b
2
2 3 (−λ 3 e 3 + µ 3 )( b 3 ) (λ 3 e 3 + µ 3 ) ···
....................................................

n
Now remove the fractions from the elements of the determinant by mul-
tiplying column j by 2 j−1 ,1 ≤ j ≤ n, and compensate for the change in
the value of the determinant by multiplying the left side by
2 1+2+3···+(n−1) =2 n(n−1)/2 .
The result is
2 n(n−1)/2 (e 1 e 2 ··· e n ) 1/2 W = |α ij e i + β ij | n , (6.7.46)
where
α ij =(−b i ) j−1 λ i ,
j−1
β ij = b µ i . (6.7.47)
i
The determinants |α ij | n , |β ij | n are both Vandermondians. Denote them by
U n and V n , respectively, and use the notation of Section 4.1.2:

U n = |α ij | n =(λ 1 λ 2 ··· λ n ) (−b i ) j−1 ,

n
, (6.7.48)
=(λ 1 λ 2 ··· λ n )[X n ] x i =−b i
V n = |β ij | n .
The determinant on the right-hand side of (6.7.46) is identical in form
with the determinant |a ij x i + b ij | n which appears in Section 3.5.3. Hence,
applying the theorem given there with appropriate changes in the symbols,
|α ij e i + β ij | n = U n |E ij | n ,
where
(n)
K
E ij = δ ij e i + ij (6.7.49)
(n) U n
and where K is the hybrid determinant obtained by replacing row i
ij
of U n by row j of V n . Removing common factors from the rows of the
determinant,
(n) µ j (n)
K =(λ 1 λ 2 ··· λ n ) H .
ij λ i ij y i =−x i =b i