Determinants and Their Applications in Mathematical Physics

64 4. Particular Determinants

Proof. Denote the left side of (a) bySm. Then, applying Lemma 4.6,

Sm=

Nm

∑

Jm

(

m

∏

r=1

x

r− 1

jr

)

sgn

{

Nm

Jm

}

V(x 1 ,x 2 ,...,xm)

=V(x 1 ,x 2 ,...,xm)

Nm

∑

Jm

sgn

{

Nm

Jm

}m

∏

r=1

x

r− 1

jr

The proof of (a) follows from Lemma 4.5. The proof of (b) follows by

applying the substitution operation

{

Nm

Jm

}

to both sides of (a).

This theorem is applied in Section 6.10.4 on the Einstein and Ernst

equations.

4.2 Symmetric Determinants

IfA=|aij|n, whereaji=aij, thenAis symmetric about its principal

diagonal. By simple reasoning,

Aji=Aij,

Ajs,ir=Air,js,

etc. Ifan+1−j,n+1−i=aij, thenAis symmetric about its secondary diago-

nal. Only the first type of determinant is normally referred to as symmetric,

but the second type can be transformed into the first type by rotation

through 90
◦
in either the clockwise or anticlockwise directions. This oper-

ation introduces the factor (−1)
n(n−1)/ 2
, that is, there is a change of sign

ifn=4m+ 2 and 4m+3,m=0, 1 , 2 ,....

Theorem.IfAis symmetric,

∑

ep{p,q,r}

Apq,rs=0,

where the symbolep{p, q, r}denotes that the sum is carried out over all

even permutations of{p, q, r}, including the identity permutation.

In this simple case the even permutations are also the cyclic permutations

[Appendix A.2].

Proof. Denote the sum by S. Then, applying the Jacobi identity

(Section 3.6.1),

AS=AApq,rs+AAqr,ps+AArp,qs

=

∣

∣

∣

∣

Apr Aps

Aqr Aqs

∣

∣

∣

∣

+

∣

∣

∣

∣

Aqp Aqs

Arp Ars

∣

∣

∣

∣

+

∣

∣

∣

∣

Arq Ars

Apq Aps

∣

∣

∣

∣