4.3 Skew-Symmetric Determinants 77
−
a 46
∣
∣
a 12 a 13 a 15
a 23 a 25
a 35
∣ ∣ ∣ ∣ ∣ ∣
+
a 56
∣
∣
a 12 a 13 a 14
a 23 a 24
a 34
∣ ∣ ∣ ∣ ∣ ∣ =
5
∑
r=1
(−1)
r+1
ar 6 Pf
(3)
r
, (4.3.26)
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 bya 26 ,isnotobtained from Pf 3 by deleting a particular row
and a particular column. It is obtained from Pf 3 by deletingallelements
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 Pfnthat one of the terms in its expansion
is
+a 12 a 34 a 56 ···a 2 n− 1 , 2 n (4.3.27)
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
Pfn=A
1 / 2
2 n
Pf
(n)
i =
[
A
(2n−1)
ii
] 1 / 2
. (4.3.28)
Skew-symmetric determinants and Pfaffians appear in Section 5.2 on the
generalized Cusick identities.
Exercises
- Theorem (Muir and Metzler)An arbitrary determinantAn=|aij|n
can be expressed as a Pfaffian of the same order.
Prove this theorem in the particular case in whichn= 3 as follows: Let
bij=
1
2
(aij+aji)=bji,
cij=
1
2
(aij−aji)=−cji.
Then
bii=aii,
cii=0,
aij−bij=cij,
aij+cji=bij.