Page 79 - Determinants and Their Applications in Mathematical Physics

64 4. Particular Determinants

Proof. Denote the left side of (a) by S m . Then, applying Lemma 4.6,

N m m
r−1
S m = x sgn N m V (x 1 ,x 2 ,...,x m )
j r
J m r=1 J m
N m m
r−1
= V (x 1 ,x 2 ,...,x m ) sgn N m x .
j r
r=1
J m
J m
The proof of (a) follows from Lemma 4.5. The proof of (b) follows by

applying the substitution operation N m to both sides of (a).
J m
This theorem is applied in Section 6.10.4 on the Einstein and Ernst
equations.

4.2 Symmetric Determinants

If A = |a ij | n , where a ji = a ij , then A is symmetric about its principal
diagonal. By simple reasoning,
A ji = A ij ,
A js,ir = A ir,js ,

etc. If a n+1−j,n+1−i = a ij , then A is 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
if n =4m + 2 and 4m +3, m =0, 1, 2,....
Theorem. If A is symmetric,

A pq,rs =0,
ep{p,q,r}
where the symbol ep{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 = AA pq,rs + AA qr,ps + AA rp,qs

= A pr A ps + A qp A qs + A rq A rs
A qr A qs A rp A rs A pq A ps