76 4. Particular Determinants
which is consistent with (4.3.22). Hence,
A
(2n−1)
ij
=(−1)
i+j
Pf
(n)
i
Pf
(n)
j
. (4.3.24)
Returning to (4.3.11) and referring to (4.3.19),
A 2 n=
[
2 n− 1
∑
i=1
(−1)
i+1
Pf
(n)
i
ai, 2 n
]
2 n− 1
∑
j=1
(−1)
j+1
Pf
(n)
j
aj, 2 n
=
[
2 n− 1
∑
i=1
(−1)
i+1
Pf
(n)
i
ai, 2 n
] 2
= [Pfn]
2
,
which completes the proof of the theorem.
The notation for Pfaffians consists of a triangular array of the elements
aijfor whichi<j:
Pfn=|a 12 a 13 a 14 ··· a 1 , 2 n
a 23 a 24 ··· a 2 , 2 n
a 34 ··· a 3 , 2 n
.........
a 2 n− 1 , 2 n
∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣
2 n− 1
. (4.3.25)
Pfnis a polynomial function of then(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 34 a 35
a 45
∣ ∣ ∣ ∣ ∣ ∣
−
a 26
∣
∣
a 13 a 14 a 15
a 34 a 35
a 45
∣ ∣ ∣ ∣ ∣ ∣
+
a 36
∣
∣
a 12 a 14 a 15
a 24 a 25
a 45