Determinants and Their Applications in Mathematical Physics

3.2 Second and Higher Minors and Cofactors 23

which can be abbreviated with the aid of the Kronecker delta function

[Appendix A]:

n

∑

q=1

ahqA

(n)

ij,pq

=A

(n)

ip

δhj−A

(n)

jp

δhi.

Similarly,

n

∑

r=1

ahrA

(n)

ijk,pqr

=A

(n)

ij,pq

δhk+A

(n)

jk,pq

δhi+A

(n)

ki,pq

δhj,

n

∑

s=1

ahsA

(n)

ijkm,pqrs

=A

(n)

ijk,pqr

δhm−A

(n)

jkm,pqr

δhi

+A

(n)

kmi,pqr

δhj−A

(n)

mij,pqr

δhk (3.2.11)

etc.

Exercise.Show that these expressions can be expressed as sums as follows:

n

∑

q=1

ahqA

(n)

ij,pq

=

∑

u,v

sgn

{

uv

ij

}

A

(n)

upδhv,

n

∑

r=1

ahrA

(n)

ijk,pqr

=

∑

u,v,w

sgn

{

uvw

ijk

}

A

(n)

uv,pq

δhw,

n

∑

s=1

ahsA

(n)

ijkm,pqrs

=

∑

u,v,w,x

sgn

{

uvw x

ijkm

}

A

(n)

uvw,pqrδhx,

etc., where, in each case, the sums are carried out over all possible cyclic

permutations of the lower parameters in the permutation symbols. A brief

note on cyclic permutations is given in Appendix A.2.

3.2.5 Scaled Cofactors....................

CofactorsA

(n)

ip,A

(n)

ij,pq,A

(n)

ijk,pqr

, etc., with both row and column parameters

written as subscripts have been defined in Section 3.2.2. They may conve-

niently be called simple cofactors. Scaled cofactorsA

ip

n,A

ij,pq

n ,A

ijk,pqr

n ,

etc., with row and column parameters written as superscripts are defined

as follows:

A

ip

n=

A

(n)

ip

An

,

A

ij,pq

n =

A

(n)

ij,pq

An

,

A

ijk,pqr

n =

A

(n)

ijk,pqr

An

, (3.2.12)