Determinants and Their Applications in Mathematical Physics

32 3. Intermediate Determinant Theory

Example 3.3. Let

V 2 n=

∣ ∣ ∣ ∣ ∣ ∣

Eip Fiq Giq

Eip Giq Fiq

Ojp Hjq Kjq

∣ ∣ ∣ ∣ ∣ ∣

2 n

,

where 2i+j=p+2q=2n. Then,V 2 n= 0 under each of the following

independent conditions:

i.j+p> 2 n,

ii.p>i,

iii. Hjq+Kjq=Ojq.

Proof. Case (i) follows immediately from Property (a). To prove case

(ii) perform row operations

V 2 n=

∣ ∣ ∣ ∣ ∣ ∣

Eip Fiq Giq

Oip (Giq−Fiq)(Fiq−Giq)

Ojp Hjq Kjq

∣ ∣ ∣ ∣ ∣ ∣

2 n

This determinant contains an (i+j)×pblock of zero elements. But,i+

j+p> 2 i+j=2n. Case (ii) follows.

To prove case (iii), perform column operations on the last determinant:

V 2 n=

∣ ∣ ∣ ∣ ∣ ∣

Eip (Fiq+Giq) Giq

Oip Oiq (Fiq−Giq)

Ojp Ojq Kjq

∣ ∣ ∣ ∣ ∣ ∣

2 n

This determinant contains an (i+j)×(p+q) block of zero elements.

However, since 2(i+j)> 2 nand 2(p+q)> 2 n, it follows thati+j+p+q>

2 n. 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 caser= 2 as follows:

∑

p<q

Nij,pqArs,pq=δij,rsA,

whereδij,rsis the generalized Kronecker delta function (Appendix A.1).

The proof follows from the fact that ifr=i, the sum represents a determi-

nant in which rowr=rowi, and if, in addition,s=j, then, in addition,

rows=rowj. In either case, the determinant is zero.