Determinants and Their Applications in Mathematical Physics

4.1 Alternants 61

b.Vn=V(x 2 ,x 3 ,...,xn)

n

∏

r=2

(xr−x 1 ).

c. V(xt,xt+1,...,xn)=V(xt+1,xt+2,...,xn)

n

∏

r=t+1

(xr−xt).

d.V

(n)

1 n

=(−1)

n+1

V(x 2 ,x 3 ,...,xn)=

(−1)

n+1

Vn

∏n

r=2

(xr−x 1 )

.

e. V

(n)

in

=(−1)

n+i

V(x 1 ,...,xi− 1 ,xi+1,...,xn)

=

(−1)

n+i

Vn

i∏− 1

r=1

(xi−xr)

∏n

r=i+1

(xr−xi)

,i> 1

f.If{j 1 j 2 ···jn}is a permutation of{ 12 ...n}, then

V

(

xj 1 ,xj 2 ,...,xjn

)

= sgn

{

12 ··· n

j 1 j 2 ··· jn

}

V(x 1 ,x 2 ,...,xn).

The proofs of (a) and (b) follow from the difference–product formula

in Section 4.1.2 and are elementary. A proof of (c) can be constructed as

follows. In (b), putn=m−t+ 1, then putxr=yr+t− 1 ,r=1, 2 , 3 ,..., and

change the dummy variable in the product fromrtosusing the formula

s=r+t−1. The resut is (c) expressed in different symbols. Whent=1,

(c) reverts to (b). The proofs of (d) and (e) are elementary. The proof

of (f) follows from Property (c) in Section 2.3.1 and Appendix A.2 on

permutations and their signs.

Let the minors ofVnbe denoted byMij. Then,

Mi=Min=V(x 1 ,...,xi− 1 ,xi+1,...,xn),

Mn=Mnn=Vn− 1.

Theorems.

a.

m

∏

r=1

Mr=

V(xm+1,xm+2,...,xn)V

m− 1

n

V(x 1 ,x 2 ,...,xm)

, 1 ≤m≤n− 1

b.

n

∏

r=1

Mr=V

n− 2

n

c.

m

∏

r=1

Mkr=

V(xk

m+1

,xk

m+2

,...,xk

n

)V

m− 1

n

V(xk 1 ,xk 2 ,...,xkm)

Proof. Use the method of induction to prove (a), which is clearly valid

whenm= 1. Assume it is valid whenm=s. Then, from Lemma (e) and