250 6. Applications of Determinants in Mathematical Physics
(A
ij
)
′
=−
∑
r
e
cru
A
rj
∑
s
e
csu
A
is
, (6.4.4)
2
∑
r
brcrA
rr
+
∑
r,s
e
(cr+cs)u
A
rs
=2
∑
r
cr, (6.4.5)
2
∑
r
brcrA
ir
A
rj
+
∑
r
e
cru
A
rj
∑
s
e
csu
A
is
=(ci+cj)A
ij
. (6.4.6)
Put
φi=
∑
s
e
csu
A
is
. (6.4.7)
Then (6.4.4) and (6.4.6) become
(A
ij
)
′
=−φiφj, (6.4.8)
2
∑
r
brcrA
ir
A
rj
+φiφj=(ci+cj)A
ij
. (6.4.9)
Eliminating theφiφjterms,
(A
ij
)
′
+(ci+cj)A
ij
=2
∑
r
brcrA
ir
A
rj
,
[
e
(ci+cj)u
A
ij
]′
=2e
(ci+cj)u
∑
r
brcrA
ir
A
rj
. (6.4.10)
Differentiating (6.4.3),
(logA)
′′
=
∑
i,j
[
e
(ci+cj)u
A
ij
]′
=2
∑
r
brcr
∑
i
e
ciu
A
ir
∑
j
e
cju
A
rj
=2
∑
r
brcrφ
2
r
. (6.4.11)
Replacingsbyrin (6.4.7),
e
ciu
φi=
∑
r
e
(ci+cr)u
A
ir
,
(
e
cju
φi
)′
=2
∑
r
brcr
(
e
ciu
A
ir
)∑
j
e
cju
A
rj
=2
∑
r
brcrφre
ciu
A
ir
,
φ
′
i+ciφi=2
∑
r
brcrφrA
ir
.
Interchangeiandr, multiply bybrcrA
rj
, sum overr, and refer to (6.4.9):
∑
r
brcrA
rj
(φ
′
r
+crφr)=2
∑
i
biciφi
∑
r
brcrA
ir
A
rj