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),

A=

n

∑

i 1 =1

n

∑

i 2 =1

ai

1 j 1

ai

2 j 2

∂

2

A

∂ai

1 j 1

∂ai

2 j 2

=

n

∑

i 1 =1

n

∑

i 2 =1

ai 1 j 1 ai 2 j 2 Ai 1 i 2 ;j 1 j 2 ,i 1 <i 2 andj 1 <j 2. (3.3.10)

Continuing in this way and applying (3.3.7) in reverse,

A=

n

∑

i 1 =1

n

∑

i 2 =1

···

n

∑

ir=1

ai

1 j 1

ai

2 j 2

···ai

rjr

∂

r

A

∂ai

1 j 1

∂ai

2 j 2

···∂ai

rjr

=

n

∑

i 1 =1

n

∑

i 2 =1

···

n

∑

ir=1

ai 1 j 1 ai 2 j 2 ···airjrAi 1 i 2 ...ir;j 1 j 2 ...jr, (3.3.11)

subject to the inequalities associated with (3.3.7) which require that theis

andjsshall be in ascending order of magnitude.

In this multiple sum, thoserth 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. Sincerdistinct dummies can

be arranged in a linear sequence inr! ways, the number of groups isr!.

Hence,

A=

(r! terms)

∑

Gk 1 k 2 ...,kr,

where

Gk 1 k 2 ...kr=

∑

i≤ik 1 <ik 2 <···<ikr≤n

aik

1

jk

1

aik

2

jk

2

···aik

rjkr

Aik

1

ik

2

···ikr;jk

1

jk

2

...jkr. (3.3.12)

In one of theser! terms, the dummiesi 1 ,i 2 ,...,irare in ascending order

of magnitude, that is,is<is+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

ofrth cofactors which concern changes in sign,

A=

∑

1 ≤i 1 <i 2 <···<ir≤n

[∑

σrai 1 j 1 ai 2 j 2 ···airjr

]

Ai 1 i 2 ...ir;j 1 j 2 ...jr,

where

σr= sgn

{

123 ··· r

i 1 i 2 i 3 ··· ir

}

. (3.3.13)

(Appendix A.2). But,

∑

σrai 1 j 1 ai 2 j 2 ···airjr=Ni 1 i 2 ...ir;j 1 j 2 ...jr.