Page 85 - Determinants and Their Applications in Mathematical Physics

70 4. Particular Determinants

In detail,
−1 • 1 1 ··· 1 1

−1 −1 • 1 ··· 1 1

−1 −1 −1 • ··· 1 1

.
B n =
.................................
−1 −1 −1 −1 ··· −1 •

−1 −1 −1 −1 ··· −1 −1

n
Lemma 4.13.
B n =(−1) .
n
Proof. Perform the column operation

C = C 2 − C 1
2
and then expand the resulting determinant by elements from the new C 2 .
The result is
B n = −B n−1 = B n−2 = ··· =(−1) n−1 B 1 .

But B 1 = −1. The result follows.
Lemma 4.14.

2n
j+k+1
a. (−1) =0,
k=1
i−1
j+k+1
b. (−1) =(−1) δ i,even ,
j
k=1
2n
j+k+1 j+1
c. (−1) =(−1) δ i,even ,
k=i
where the δ functions are defined in Appendix A.1. All three identities follow
from the elementary identity
q

(−1) =(−1) δ q−p,even .
k
p
k=p
Define the function E ij as follows:
 i+j+1
 (−1) , i < j
E ij = 0, i = j
−(−1) , i>j.
 i+j+1
Lemma 4.15.

2n
j+1
a. E jk =(−1) ,
k=1