Page 102 - Determinants and Their Applications in Mathematical Physics

4.5 Centrosymmetric Determinants 87

a 1 + a 5 a 2 + a 4 2a 3 • •

b 1 + b 5 b 2 + b 4 2b 3 • •

c 1 c 2 c 3 • •
=

b 5 b 4 b 3 b 2 − b 4
b 1 − b 5

a 5 a 4 a 3 a 2 − a 4 a 1 − a 5

a 1 + a 5 a 2 + a 4

b 2 − b 4 b 1 − b 5
2a 3

= b 1 + b 5 b 2 + b 4 2b 3
a 2 − a 4 a 1 − a 5

c 1 c 2 c 3
1
= |E||F|, (4.5.5)
2
where
   
a 1 a 2 a 3 a 5 a 4 a 3
E =  b 1 b 2 b 3  +  b 5 b 4 b 3  ,
c 1 c 2 c 3 c 1 c 2 c 3

b 2 b 1 b 4 b 5
F = − . (4.5.6)
a 2 a 1 a 4 a 5
Two of these matrices are submatrices of A 5 . The other two are submatrices
with their rows or columns arranged in reverse order.
Exercise. If a determinant A n is symmetric about its principal diagonal
and persymmetric (Hankel, Section 4.8) about its secondary diagonal, prove
analytically that A n is centrosymmetric.
4.5.2 Symmetric Toeplitz Determinants
The classical Toeplitz determinant A n is defined as follows:
A n = |a i−j | n

a 0 a −1 a −2 a −3 ··· a −(n−1)

a 1 a 0 a −1 a −2 ··· ···

a 2 a 1 a 0 a −1 ··· ···
.
=
a 3 a 2 a 1 a 0 ··· ···

··· ··· ··· ··· ··· ···

a n−1 ··· ··· ··· ··· a 0
n
The symmetric Toeplitz determinant T n is defined as follows:
T n = |t |i−j| | n

t 0 t 1 t 2 t 3 ··· t n−1

t 1 t 0 t 1 t 2 ··· ···

t 2 t 1 t 0 t 1 ··· ···
, (4.5.7)
=
t 3 t 2 t 1 t 0 ··· ···

··· ··· ··· ··· ··· ···

t n−1 ··· ··· ··· ··· t 0
n