Page 293 - Determinants and Their Applications in Mathematical Physics

278 6. Applications of Determinants in Mathematical Physics

Applying (A),

rr
v = D x (log A)= − λ r e r A , (6.8.3)
r

D y (log A)= λ r µ r e r A , (6.8.4)
rr
r
2 2
D t (log A)=4 λ r (b − b r c r + c )e r A . (6.8.5)
rr
r r
r
Applying (B),

D x (A )= λ r e r A A , (6.8.6)
rj
ir
ij
r

rj
D y (A )= − λ r µ r e r A A , (6.8.7)
ir
ij
r
2 2
D t (A )= −4 λ r (b − b r c r + c )e r A A . (6.8.8)
ij
ir
rj
r r
r
Applying (C) with
i. f r = b r , g r = c r ;
2
2
ii. f r = b , g r = −c ;
r r
3
3
iii. f r = b , g r = c ;
r r
in turn,

λ r e r A rr + A rs = λ r , (6.8.9)
r r,s r

λ r µ r e r A rr + (b r − c s )A rs = λ r µ r , (6.8.10)
r r,s r
2 2 2 2
λ r (b − b r c r + c )e r A rr + (b − b r c s + c )A rs
r r r s
r r,s
2 2
= λ r (b − b r c r + c ). (6.8.11)
r r
r
Applying (D) with (i)–(iii) in turn,

λ r e r A A rj + A A rj =(b i + c j )A , (6.8.12)
is
ij
ir
r r,s
2 2
λ r µ r e r A A rj + (b r − c s )A A rj =(b − c )A , (6.8.13)
is
ir
ij
i j
r r,s
2 2 2 2
λ r (b − b r c r + c )e r A A rj + (b − b r c s + c )A A rj
ir
is
r r r s
r r,s
3
3
=(b + c )A . (6.8.14)
ij
i j
Eliminating the sum common to (6.8.3) and (6.8.9), the sum common to
(6.8.4) and (6.8.10) and the sum common to (6.8.5) and (6.8.11), we find