Determinants and Their Applications in Mathematical Physics

310 Appendix

=(−1)

i+m+1

sgn

{

12 ... (i−1)(i+1) ... (m−1)

r 3 r 4 ... ... ... rm

}

m− 2

,

where 1 ≤rk≤m−2,rk=i, and 3≤k≤m.

Proof. The casesi= 1 andi>1 are considered separately. Wheni=1,

then 2≤rk≤m−1. Letpdenote the number of inversions required to

transform the set{r 3 r 4 ...rm}m− 2 into the set{ 23 ...(m−1)}m− 2 , that

is,

(−1)

p

= sgn

{

23 ... (m−1)

r 3 r 4 ... rm

}

m− 2

.

Hence

sgn

{

12 3 4... m

imr 3 r 4 ... rm

}

m

=(−1)

p

sgn

{

1234 ... m

im 23 ... (m−1)

}

m

=(−1)

p+m− 2

sgn

{

1234 ... (m−1)m

1234 ... (m−1)m

}

m

=(−1)

p+m− 2

=(−1)

m− 2

sgn

{

23 ... (m−1)

r 3 r 4 ... rm

}

m− 2

,

which proves the lemma wheni=1.

Wheni>1, letqdenote the number of inversions required to transform

the set{r 3 r 4 ···rm}m− 2 into the set{ 12 ···(i−1)(i+1)···(m−1)}m− 2.

Then,

(−1)

q

= sgn

{

12 ... (i−1)(i+1) ... (m−1)

r 3 r 4 ... ... ... rm

}

m− 2

.

Hence,

sgn

{

12 3 4··· m

imr 3 r 4 ··· rm

}

m

=(−1)

q

sgn

{

1234 ··· ··· ··· (m−1) m

im 12 ··· (i−1)(i+1) ··· (m−2) (m−1)

}

m

=(−1)

q+m

sgn

{

1234 ··· ··· ··· (m−1) m

i 123 ··· (i−1)(i+1) ··· (m−1) m

}

m

=(−1)

q+m+i− 1

sgn

{

1234 ··· m

1234 ··· m

}

m

=(−1)

q+m+i− 1

=(−1)

m+i− 1

sgn

{

12 ··· (i−1)(i+1) ··· (m−1)

r 3 r 4 ··· ··· ··· rm

}

m− 2

,