Determinants and Their Applications in Mathematical Physics

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

∣ ∣ ∣ ∣ ∣ ∣