Page 100 - Determinants and Their Applications in Mathematical Physics

4.5 Centrosymmetric Determinants 85

Exercises
1. Prove that when n = 3 and (x 1 ,x 2 ) → (x, y),
∂
[H 1 ,H 2 ,H 3 ]=[H 2 ,H 3 ,H 1 ],
∂x
∂
[H 1 ,H 2 ,H 3 ]=[H 3 ,H 1 ,H 2 ]
∂y
and apply these formulas to give an alternative proof of the particular
circulant identity
A(H 1 ,H 2 ,H 3 )=1.
If y = 0, prove that

∞
x 3r
H 1 = ,
(3r)!
r=0
∞ 3r+2
x
H 2 = ,
(3r + 2)!
r=0
∞ 3r+1
x
H 3 = .
(3r + 1)!
r=0
2. Apply the partial derivative method to give an alternative proof of the
general circulant identity as stated in the theorem.

4.5 Centrosymmetric Determinants

4.5.1 Definition and Factorization
The determinant A n = |a ij | n , in which
(4.5.1)
a n+1−i,n+1−j = a ij
is said to be centrosymmetric. The elements in row (n+1−i) are identical
with those in row i but in reverse order; that is, if

R i = a i1 a i2 ...a i,n−1 a in ,
then

R n+1−i = a in a i,n−1 ...a i2 a i1 .
A similar remark applies to columns. A n is unaltered in form if it is trans-
posed first across one diagonal and then across the other, an operation
◦
which is equivalent to rotating A n in its plane through 180 in either di-
rection. A n is not necessarily symmetric across either of its diagonals. The