Determinants and Their Applications in Mathematical Physics

(Chris Devlin) #1
6.10 The Einstein and Ernst Equations 297

Proof. Proof of (a).Denote the determinant on the left byWm.


wi+j− 2 +wi+j=

2 n

k=1

ykx

i+j− 2
k

,

where


yk=(−1)

k+1
εkMk(c). (6.10.33)

Hence, applying the lemma in Section 4.1.7 withN→ 2 nandn→m,


Wm=






2 n

k=1

ykx

i+j− 2
k

∣ ∣ ∣ ∣ ∣ =

2 n

k 1 ,k 2 ,...,km=1

Ym

(

m

r=2

x

r− 1
kr

)



x

j− 1
ki



m

,

where


Ym=

m

r=1

yk
r

. (6.10.34)

Hence, applying Identity 4 in Appendix A.3,


Wm=

1

m!

2 n

k 1 ,k 2 ,...,km=1

Ym

k 1 ,k 2 ,...km

j 1 ,j 2 ,...,jm

(

m

r=2

x

r− 1
jr

)

V(xj 1 ,xj 2 ,...,xjm).

(6.10.35)

Applying Theorem (b) in Section 4.1.9 on Vandermondian identities,

Wm=

1

m!

2 n

k 1 ,k 2 ,...,km=1

Ym

{

V(xk
1
,xk
2
,...,xk
m

)

} 2

. (6.10.36)

Due to the presence of the squared Vandermondian factor, the conditions of


Identity 3 in Appendix A.3 withN→ 2 nare satisfied. Also, eliminating the


x’s using (6.10.26) and (6.10.28) and referring to Exercise 3 in Section 4.1.2,


{

(V(xk
1
,xk
2
,...,xk
m

)

} 2


−m(m−1)

{

V(ck
1
,ck
2
,...,ck
m

)

} 2

. (6.10.37)

Hence,


Wm=ρ

−m(m−1)


1 ≤k 1 <k 2 <...<km≤ 2 n

Ym

{

V(ck 1 ,ck 2 ,...,ckm)

} 2

. (6.10.38)

From (6.10.33) and (6.10.34),


Ym=(−1)

K
Em

m

r=1

Mkr(c),
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