Page 92 - Determinants and Their Applications in Mathematical Physics

4.3 Skew-Symmetric Determinants 77

a 46 a 12 a 13 a 15 a 56 a 12 a 13 a 14

− a 23 a 25 + a 23 a 24
a 35 a 34
5
r+1 (3)
= (−1) a r6 Pf , (4.3.26)
r
r=1
which illustrates (4.3.19). This formula can be regarded as an expansion
of Pf 3 by the five elements from the fifth column and their associated
second-order Pfaffians. Note that the second of these five Pfaffians, which
is multiplied by a 26 ,is not obtained from Pf 3 by deleting a particular row
and a particular column. It is obtained from Pf 3 by deleting all elements
whose suffixes include either 2 or 6 whether they be row parameters or
column parameters. The other four second-order Pfaffians are obtained in
a similar manner.
It follows from the definition of Pf n that one of the terms in its expansion
is

(4.3.27)
+ a 12 a 34 a 56 ··· a 2n−1,2n
in which the parameters are in ascending order of magnitude. This term is
known as the principal term. Hence, there is no ambiguity in signs in the
relations
1/2
Pf n = A
2n
(n) (2n−1) 1/2

Pf = A . (4.3.28)
i ii
Skew-symmetric determinants and Pfaffians appear in Section 5.2 on the
generalized Cusick identities.

Exercises

1. Theorem (Muir and Metzler) An arbitrary determinant A n = |a ij | n
can be expressed as a Pfaffian of the same order.
Prove this theorem in the particular case in which n = 3 as follows: Let

1
b ij = (a ij + a ji )= b ji ,
2
1
c ij = (a ij − a ji )= −c ji .
2
Then
b ii = a ii ,
c ii =0,
a ij − b ij = c ij ,
a ij + c ji = b ij .