Determinants and Their Applications in Mathematical Physics

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

1. 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.