Page 47 - Determinants and Their Applications in Mathematical Physics

32 3. Intermediate Determinant Theory

Example 3.3. Let

E ip F iq
G iq
,
G iq
V 2n = E ip
F iq
O jp H jq K jq
2n
where 2i + j = p +2q =2n. Then, V 2n = 0 under each of the following
independent conditions:
i. j + p> 2n,
ii. p>i,
iii. H jq + K jq = O jq .

Proof. Case (i) follows immediately from Property (a). To prove case
(ii) perform row operations

E ip
F iq G iq
(G iq − F iq )(F iq − G iq ) .

V 2n = O ip

O jp H jq K jq
2n
This determinant contains an (i + j) × p block of zero elements. But, i +
j + p> 2i + j =2n. Case (ii) follows.
To prove case (iii), perform column operations on the last determinant:

(F iq + G iq )

E ip G iq
(F iq − G iq ) .
V 2n = O ip
O iq

K jq
O jq
O jp
2n
This determinant contains an (i + j) × (p + q) block of zero elements.
However, since 2(i+j) > 2n and 2(p+q) > 2n, it follows that i+j+p+q>
2n. Case (iii) follows.
3.3.4 The Laplace Sum Formula
The simple sum formula for elements and their cofactors (Section 2.3.4),
which incorporates the theorem on alien cofactors, can be generalized for
the case r = 2 as follows:

N ij,pq A rs,pq = δ ij,rs A,
p<q
where δ ij,rs is the generalized Kronecker delta function (Appendix A.1).
The proof follows from the fact that if r = i, the sum represents a determi-
nant in which row r =row i, and if, in addition, s = j, then, in addition,
row s =row j. In either case, the determinant is zero.