Page 43 - Determinants and Their Applications in Mathematical Physics

28 3. Intermediate Determinant Theory

Substituting the first line of (3.3.9 and the second line of (3.3.8),
n n 2
∂ A
A = a i 1 j 1 i 2 j 2
a
∂a i 1 j 1 ∂a i 2 j 2
i 1 =1 i 2 =1
n n

a
= a i 1 j 1 i 2 j 2 A i 1 i 2 ;j 1 j 2 , i 1 <i 2 and j 1 <j 2 . (3.3.10)
i 1 =1 i 2 =1
Continuing in this way and applying (3.3.7) in reverse,
n n n
r
∂ A
A = ··· a i 1 j 1 i 2 j 2 ··· a i r j r
a
∂a i 1 j 1 ∂a i 2 j 2 ··· ∂a i r j r
i 1 =1 i 2 =1 i r =1
n n n

= ··· a i 1 j 1 i 2 j 2 ··· a i r j r A i 1 i 2 ...i r ;j 1 j 2 ...j r , (3.3.11)
a
i 1 =1 i 2 =1 i r =1
subject to the inequalities associated with (3.3.7) which require that the i s
and j s shall be in ascending order of magnitude.
In this multiple sum, those rth cofactors in which the dummy variables
are not distinct are zero so that the corresponding terms in the sum are
zero. The remaining terms can be divided into a number of groups according
to the relative magnitudes of the dummies. Since r distinct dummies can
be arranged in a linear sequence in r! ways, the number of groups is r!.
Hence,
(r! terms)

A = G k 1 k 2 ...,k r ,
where

=
G k 1 k 2 ...k r a i k 1 k 1 a i k 2 k 2
j j
i≤i k 1 <i k 2 <···<i k r ≤n
. (3.3.12)
j A i k 1 k 2
··· a i k r k r i ···i k r ;j k 1 k 2 ...j k r
j
In one of these r! terms, the dummies i 1 ,i 2 ,...,i r are in ascending order
of magnitude, that is, i s <i s+1 ,1 ≤ s ≤ r − 1. However, the dummies
in the other (r! − 1) terms can be interchanged in such a way that the
inequalities are valid for those terms too. Hence, applying those properties
of rth cofactors which concern changes in sign,

a
A = σ r a i 1 j 1 i 2 j 2 ··· a i r j r A i 1 i 2 ...i r ;j 1 j 2 ...j r ,
1≤i 1 <i 2 <···<i r ≤n
where

1 2 3 ··· r
σ r = sgn . (3.3.13)
i 1 i 2 i 3 ··· i r
(Appendix A.2). But,

a .
σ r a i 1 j 1 i 2 j 2 ··· a i r j r = N i 1 i 2 ...i r ;j 1 j 2 ...j r