Determinants and Their Applications in Mathematical Physics

272 6. Applications of Determinants in Mathematical Physics

Proof.

D

j

x

(φi)=

(

1

2

bi

)j

e

− 1 / 2

i

[(−1)

j

λiei+μi]

so that every element in rowiofW contains the factore

− 1 / 2

i. Removing

all these factors from the determinant,

(e 1 e 2 ...en)

1 / 2

W

=

∣ ∣ ∣ ∣ ∣ ∣ ∣

λ 1 e 1 +μ 1

1

2

b 1 (−λ 1 e 1 +μ 1 )(

1

2

b 1 )

2

(λ 1 e 1 +μ 1 ) ···

λ 2 e 2 +μ 2

1

2

b 2 (−λ 2 e 2 +μ 2 )(

1

2

b 2 )

2

(λ 2 e 2 +μ 2 ) ···

λ 3 e 3 +μ 3

1

2

b 3 (−λ 3 e 3 +μ 3 )(

1

2

b 3 )

2

(λ 3 e 3 +μ 3 ) ···

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

∣ ∣ ∣ ∣ ∣ ∣ ∣ n

(6.7.45)

Now remove the fractions from the elements of the determinant by mul-

tiplying columnjby 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 ···en)

1 / 2

W=|αijei+βij|n, (6.7.46)

where

αij=(−bi)

j− 1

λi,

βij=b

j− 1

i μi. (6.7.47)

The determinants|αij|n,|βij|nare both Vandermondians. Denote them by

UnandVn, respectively, and use the notation of Section 4.1.2:

Un=|αij|n=(λ 1 λ 2 ···λn)

∣

∣(−b

i)

j− 1

∣

∣

n

,

=(λ 1 λ 2 ···λn)[Xn]x

i=−bi

, (6.7.48)

Vn=|βij|n.

The determinant on the right-hand side of (6.7.46) is identical in form

with the determinant|aijxi+bij|nwhich appears in Section 3.5.3. Hence,

applying the theorem given there with appropriate changes in the symbols,

|αijei+βij|n=Un|Eij|n,

where

Eij=δijei+

K

(n)

ij

Un

(6.7.49)

and whereK

(n)

ij is the hybrid determinant obtained by replacing rowi

ofUnby rowj ofVn. Removing common factors from the rows of the

determinant,

K

(n)

ij

=(λ 1 λ 2 ···λn)

μj

λi

[

H

(n)

ij

]

yi=−xi=bi

.