Determinants and Their Applications in Mathematical Physics

56 4. Particular Determinants

This completes the proof of Theorem 4.3 which can be expressed in the

form

Hrs

Xn

=

∏n

i=1

(ys−xi)

(ys−xr)

∏n

i=1

i=r

(xr−xi)

.

Let

An=|σ

(m)

i,j− 1

|n

=

∣

∣

∣

∣

∣

∣

∣

∣

σ

(m)

10

σ

(m)

11

... σ

(m)

1 ,n− 1

σ

(m)

20 σ

(m)

21 ... σ

(m)

2 ,n− 1

........................

σ

(m)

n 0 σ

(m)

n 1 ... σ

(m)

n,n− 1

∣

∣

∣

∣

∣

∣

∣

∣

n

,m≥n.

Theorem 4.4.

An=(−1)

n(n−1)/ 2

Xn.

Proof.

An=

∣

∣C

0 C 1 C 2 ...Cn− 1 ,

∣

∣

n

where, from the lemma in Appendix A.7,

Cj=

[

σ

(m)

1 j

σ

(m)

2 j

σ

(m)

3 j

... σ

(m)

nj

]T

=

j

∑

p=0

σ

(m)

p

[

v

j−p

1 v

j−p

2 v

j−p

3 ... v

j−p

n

]T

,vr=−xr,σ

(m)

0 =1.

Applying the column operations

C

′

j

=Cj−

j

∑

k=1

σ

(m)

k

Cj−k

in the orderj=1, 2 , 3 ,...so that each new column created by one operation

is applied in the next operation, it is found that

C

′

j

=

[

v

j

1

v

j

2

v

j

3

... v

j

n

]T

,j=0, 1 , 2 ,....

Hence

An=|v

j− 1

i

|n

=(−1)

n(n−1)/ 2

|x

j− 1

i

|n.

Theorem 4.4 follows.