Page 201 - Determinants and Their Applications in Mathematical Physics

186 5. Further Determinant Theory
 
2n−1 2n−1

i+1 (n) j+1 (n)
= − (−1) Pf  (−1) Pf 
i a i,2n j φ j
i=1 j=1
= −Pf n H n K n−1 . (5.2.27)
Part (c) now follows from Theorem 5.1 and (d) is proved in a similar
manner.

Let R(φ) denote the row vector defined as

R(φ)= φ 1 φ 2 φ 3 ··· φ 2n−1 •
and let B 2n (φ, ψ) denote the determinant of order 2n which is obtained
from |a ij | 2n by replacing the last row by −R(φ) and replacing the last
column by R (ψ).
T
Theorem 5.7.
B 2n (φ, ψ)= H n−1 H n K n−1 K n .
Proof.
2n−1 2n−1
(2n−1)
B 2n (φ, ψ)= ψ i φ j A .
ij
i=1 j=1
The theorem now follows (5.2.13), (5.2.24), and (5.2.25).
Theorem 5.8.
(2n)
B 2n (φ, ψ)= A .
2n−1,2n
Proof. Applying the Jacobi identity (Section 3.6),

(2n) (2n)
A 2n−1,2n−1 A 2n−1,2n (2n)

(2n) (2n) = A 2n A 2n−1,2n;2n−1,2n . (5.2.28)
A A

2n,2n−1 2n,2n
(2n)
But, A , i =2n − 1, 2n, are skew-symmetric of odd order and are
ii
therefore zero. The other two first cofactors are equal in magnitude but
opposite in sign. Hence,
(2n) 2
A = A 2n A 2n−2 ,
2n−1,2n
(2n)
A =Pf n Pf n−1 . (5.2.29)
2n−1,2n
Theorem 5.8 now follows from Theorems 5.1 and 5.7.
If ψ r = φ r , then K n = H n and Theorems 5.1, 5.6a and c, and 5.7
degenerate into identities published in a different notation by Cusick,
namely,
4
A 2n = H ,
n
B 2n−1 (φ)= H 3 H n ,
n−1