Determinants and Their Applications in Mathematical Physics

186 5. Further Determinant Theory

=−

[

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 φj





=−PfnHnKn− 1. (5.2.27)

Part (c) now follows from Theorem 5.1 and (d) is proved in a similar

manner.

LetR(φ) denote the row vector defined as

R(φ)=

[

φ 1 φ 2 φ 3 ···φ 2 n− 1 •

]

and letB 2 n(φ, ψ) denote the determinant of order 2nwhich is obtained

from|aij| 2 nby replacing the last row by−R(φ) and replacing the last

column byR

T

(ψ).

Theorem 5.7.

B 2 n(φ, ψ)=Hn− 1 HnKn− 1 Kn.

Proof.

B 2 n(φ, ψ)=

2 n− 1

∑

i=1

2 n− 1

∑

j=1

ψiφjA

(2n−1)

ij

.

The theorem now follows (5.2.13), (5.2.24), and (5.2.25).

Theorem 5.8.

B 2 n(φ, ψ)=A

(2n)

2 n− 1 , 2 n.

Proof. Applying the Jacobi identity (Section 3.6),

∣

∣

∣

∣

∣

A

(2n)

2 n− 1 , 2 n− 1 A

(2n)

2 n− 1 , 2 n

A

(2n)

2 n, 2 n− 1

A

(2n)

2 n, 2 n

∣

∣

∣

∣

∣

=A 2 nA

(2n)

2 n− 1 , 2 n;2n− 1 , 2 n

. (5.2.28)

But,A

(2n)

ii ,i =2n−1, 2n, are skew-symmetric of odd order and are

therefore zero. The other two first cofactors are equal in magnitude but

opposite in sign. Hence,

(

A

(2n)

2 n− 1 , 2 n

) 2

=A 2 nA 2 n− 2 ,

A

(2n)

2 n− 1 , 2 n

=PfnPfn− 1. (5.2.29)

Theorem 5.8 now follows from Theorems 5.1 and 5.7.

Ifψr =φr, thenKn =Hn and Theorems 5.1, 5.6a and c, and 5.7

degenerate into identities published in a different notation by Cusick,

namely,

A 2 n=H

4

n

,

B 2 n− 1 (φ)=H

3

n− 1

Hn,