Page 263 - Determinants and Their Applications in Mathematical Physics

248 6. Applications of Determinants in Mathematical Physics

Applying double-sum identities (C) and (A),
n n n
r+s−1
[x +(−1) r+s c]A rs = (2r − 1)
r=1 s=1 r=1
= n 2 (6.3.12)

xA r+s−1
n
n
= x A rs
A
r=1 s=1
n n

2
= n − c (−1) r+s A . (6.3.13)
rs
r=1 s=1
Differentiating and using (6.3.10),

n n
xA
= c (−1) r+s
A α r α s
r s
2
= cλ . (6.3.14)
It follows from (6.3.5) that
xA 1 2 11
= 1 − [n − (c − 1)A ]
A 1+ x
(c − 1)xA + n
11 2
2
= n − . (6.3.15)
1+ x
Hence, eliminating xA /A and using (6.3.14),

11 2
(c − 1)xA + n 2
= −cλ . (6.3.16)
1+ x
Differentiating (6.3.7) and using (6.3.10) and the first equation in (6.3.8),
β = λα i . (6.3.17)

i
Differentiating the second equation in (6.3.11) and using (6.3.17),
11
(xA ) =2cλα 1 β 1 . (6.3.18)
All preparations for proving the theorem are now complete.
Put
11
y =4(c − 1)xA .
Referring to the second equation in (6.3.11),
11
y =4(c − 1)(xA )

2
=4c(c − 1)β . (6.3.19)
1
Referring to the first equation in (6.3.11),
y
1 2
11
=4(c − 1)(A )
x