4.3 Skew-Symmetric Determinants 73
=0
λii=(−1)
i+1
[
δi,odd−δi,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 Pfnis defined by the following formula, which
is similar in nature to the formula which defines the determinantAnin
Section 1.2:
Pfn=
∑
sgn
{
1234 ···(2n−1) 2n
i 1 j 1 i 2 2 ··· in jn
}
2 n
ai 1 j 1 ai 2 j 2 ···ainjn,
(4.3.13)
where the sum extends over all possible distinct terms subject to the
restriction
1 ≤is<js≤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
n
∏
s=1
(2s−1) =
(2n)!
2
n
n!
. (4.3.15)
Illustrations
Pf 1 =
∑
sgn
{
12
ij
}
aij (1 term)
=a 12 ,
A 2 = [Pf 1 ]
2
Pf 2 =
∑
sgn
{
1234
i 1 j 1 i 2 j 2
}
ai
1 j 1
ai
2 j 2
(3 terms). (4.3.16)
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
A 4 = [Pf 2 ]
2
. (4.3.17)
These results agree with (4.3.7) and (4.3.8).