Page 91 - Determinants and Their Applications in Mathematical Physics

76 4. Particular Determinants

which is consistent with (4.3.22). Hence,
(2n−1) (n) (n)
A =(−1) i+j Pf Pf . (4.3.24)
ij i j
Returning to (4.3.11) and referring to (4.3.19),
 
2n−1 2n−1

i+1 (n) j+1 (n)
A 2n = (−1) Pf  (−1) Pf 
i a i,2n j a j,2n
i=1 j=1
2
2n−1

i+1 (n)
= (−1) Pf
i a i,2n
i=1
2
= [Pf n ] ,
which completes the proof of the theorem.
The notation for Pfaffians consists of a triangular array of the elements
a ij for which i<j:

Pf n = |a 12 a 13 a 14 ··· a 1,2n

a 23 a 24 ··· a 2,2n

a 34 ··· a 3,2n . (4.3.25)
.........

2n−1
a 2n−1,2n
Pf n is a polynomial function of the n(2n − 1) elements in the array.
Illustrations
From (4.3.16), (4.3.17), and (4.3.25),
Pf 1 = |a 12 | = a 12 ,

Pf 2 = a 12 a 13 a 14

a 23 a 24
a 34
= a 12 a 34 − a 13 a 24 + a 14 a 23 .
It is left as an exercise for the reader to evaluate Pf 3 directly from the defini-
tion (4.3.13) with the aid of the notes given in the section on permutations
associated with Pfaffians in Appendix A.2 and to show that

Pf 3 = a 12 a 13 a 14 a 15 a 16

a 23 a 24 a 25 a 26

a 34 a 35 a 36

a 45 a 46

a 56

= a 16 a 23 a 24 a 25 a 26 a 13 a 14 a 15 a 36 a 12 a 14 a 15

a 34 a 35 − a 34 a 35 + a 24 a 25
a 45 a 45 a 45