Determinants and Their Applications in Mathematical Physics

4.5 Centrosymmetric Determinants 87

=

∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣

a 1 +a 5 a 2 +a 4 2 a 3 ••

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

b 1 +b 5 b 2 +b 4 2 b 3

c 1 c 2 c 3

∣ ∣ ∣ ∣ ∣ ∣

∣

∣

∣

∣

b 2 −b 4 b 1 −b 5

a 2 −a 4 a 1 −a 5

∣

∣

∣

∣

=

1

2

|E||F|, (4.5.5)

where

E=





a 1 a 2 a 3

b 1 b 2 b 3

c 1 c 2 c 3



+





a 5 a 4 a 3

b 5 b 4 b 3

c 1 c 2 c 3



,

F=

[

b 2 b 1

a 2 a 1

]

−

[

b 4 b 5

a 4 a 5

]

. (4.5.6)

Two of these matrices are submatrices ofA 5. The other two are submatrices

with their rows or columns arranged in reverse order.

Exercise.If a determinantAnis symmetric about its principal diagonal

and persymmetric (Hankel, Section 4.8) about its secondary diagonal, prove

analytically thatAnis centrosymmetric.

4.5.2 Symmetric Toeplitz Determinants..........

The classical Toeplitz determinantAnis defined as follows:

An=|ai−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 ··· ···

··· ··· ··· ··· ··· ···

an− 1 ··· ··· ··· ··· a 0

∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ n

.

The symmetric Toeplitz determinantTnis defined as follows:

Tn=|t|i−j||n

=

∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣

t 0 t 1 t 2 t 3 ··· tn− 1

t 1 t 0 t 1 t 2 ··· ···

t 2 t 1 t 0 t 1 ··· ···

t 3 t 2 t 1 t 0 ··· ···

··· ··· ··· ··· ··· ···

tn− 1 ··· ··· ··· ··· t 0

∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ n

, (4.5.7)