Page 88 - Determinants and Their Applications in Mathematical Physics

4.3 Skew-Symmetric Determinants 73

=0
λ ii =(−1) i+1 δ i,odd − δ i,even
=1.
This completes the proofs of the preparatory lemmas. The definition of a
Pfaffian follows. The above lemmas will be applied to prove the theorem
which relates it to a skew-symmetric determinant.

4.3.3 Pfaffians

The nth-order Pfaffian Pf n is defined by the following formula, which
is similar in nature to the formula which defines the determinant A n in
Section 1.2:

1 2 3 4 ··· (2n − 1) 2n

a
Pf n = sgn a i 1 j 1 i 2 j 2 ··· a i n j n ,
i 1 j 1 i 2  2 ··· i n j n
2n
(4.3.13)
where the sum extends over all possible distinct terms subject to the
restriction
1 ≤ i s <j s ≤ n, 1 ≤ s ≤ n.. (4.3.14)
Notes on the permutations associated with Pfaffians are given in
Appendix A.2. The number of terms in the sum is

(2n)!
n

(2s − 1) = . (4.3.15)
2 n!
n
s=1
Illustrations

12

Pf 1 = sgn (1 term)
i j a ij
= a 12 ,
A 2 = [Pf 1 ] 2

1 2 3 4
a
Pf 2 = sgn a i 1 j 1 i 2 j 2 (3 terms). (4.3.16)
i 1 j 1 i 2 j 2
Omitting the upper parameters,
Pf 2 = sgn{1234}a 12 a 34 + sgn{1324}a 13 a 24 + sgn{1423}a 14 a 23
= a 12 a 34 − a 13 a 24 + a 14 a 23
2
A 4 = [Pf 2 ] . (4.3.17)
These results agree with (4.3.7) and (4.3.8).