Page 38 - 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
(n) (n) (n)

a hq A = A δ hj − A δ hi .
ij,pq ip jp
q=1
Similarly,
n
(n) (n) (n) (n)
a hr A = A ij,pq hk + A jk,pq hi + A ki,pq hj ,
δ
δ
δ
ijk,pqr
r=1
n
(n) (n) (n)
a hs A = A ijk,pqr hm − A δ
δ
ijkm,pqrs jkm,pqr hi
s=1
(n) (n)
+ A kmi,pqr hj − A mij,pqr hk (3.2.11)
δ
δ
etc.
Exercise. Show that these expressions can be expressed as sums as follows:

n
(n) uv (n)
a hq A = sgn A δ hv ,
ij,pq i j up
q=1 u,v

n
(n) uv w (n)
a hr A = sgn A uv,pq hw ,
δ
ijk,pqr i j k
r=1 u,v,w

n
(n) uv w x (n)
δ
a hs A = sgn A uvw,pqr hx ,
ijkm,pqrs i j k m
s=1 u,v,w,x
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
(n) (n) (n)
Cofactors A , A , A , etc., with both row and column parameters
ip ij,pq ijk,pqr
written as subscripts have been defined in Section 3.2.2. They may conve-
niently be called simple cofactors. Scaled cofactors A , A ij,pq , A ijk,pqr ,
ip
n n n
etc., with row and column parameters written as superscripts are defined
as follows:
(n)
A
A ip = ip ,
n A n
(n)
A
A ij,pq = ij,pq ,
n A n
(n)
A
A ijk,pqr = ijk,pqr , (3.2.12)
n A n