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