Determinants and Their Applications in Mathematical Physics

4.5 Centrosymmetric Determinants 85

Exercises

1.Prove that whenn= 3 and (x 1 ,x 2 )→(x, y),

∂

∂x

[H 1 ,H 2 ,H 3 ]=[H 2 ,H 3 ,H 1 ],

∂

∂y

[H 1 ,H 2 ,H 3 ]=[H 3 ,H 1 ,H 2 ]

and apply these formulas to give an alternative proof of the particular

circulant identity

A(H 1 ,H 2 ,H 3 )=1.

Ify= 0, prove that

H 1 =

∞

∑

r=0

x

3 r

(3r)!

,

H 2 =

∞

∑

r=0

x

3 r+2

(3r+ 2)!

,

H 3 =

∞

∑

r=0

x

3 r+1

(3r+ 1)!

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 determinantAn=|aij|n, in which

an+1−i,n+1−j=aij (4.5.1)

is said to be centrosymmetric. The elements in row (n+1−i) are identical

with those in rowibut in reverse order; that is, if

Ri=

[

ai 1 ai 2 ...ai,n− 1 ain

]

,

then

Rn+1−i=

[

ainai,n− 1 ...ai 2 ai 1

]

.

A similar remark applies to columns.Anis unaltered in form if it is trans-

posed first across one diagonal and then across the other, an operation

which is equivalent to rotatingAnin its plane through 180
◦
in either di-

rection.Anis not necessarily symmetric across either of its diagonals. The