Determinants and Their Applications in Mathematical Physics

86 4. Particular Determinants

most general centrosymmetric determinant of order 5 is of the form

A 5 =

∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣

a 1 a 2 a 3 a 4 a 5

b 1 b 2 b 3 b 4 b 5

c 1 c 2 c 3 c 2 c 1

b 5 b 4 b 3 b 2 b 1

a 5 a 4 a 3 a 2 a 1

∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣

. (4.5.2)

Theorem.Every centrosymmetric determinant can be factorized into two

determinants of lower order.A 2 nhas factors each of order n, whereas

A 2 n+1has factors of ordersnandn+1.

Proof. In the row vector

Ri+Rn+1−i=

[

(ai 1 +ain)(ai 2 +ai,n− 1 )···(ai,n− 1 +ai 2 )(ain+ai 1 )

]

,

the (n+1−j)th element is identical to thejth element. This suggests

performing the row and column operations

R

′

i

=Ri+Rn+1−i, 1 ≤i≤

[

1

2

n

]

,

C

′

j

=Cj−Cn+1−j,

[

1

2

(n+1)

]

+1≤j≤n,

where

[

1

2

n

]

is the integer part of

1

2

n. The result of these operations is

that an array of zero elements appears in the top right-hand corner ofAn,

which then factorizes by applying a Laplace expansion (Section 3.3). The

dimensions of the various arrays which appear can be shown clearly using

the notationMrs, etc., for a matrix withrrows andscolumns. (^0) rsis an
array of zero elements.
A 2 n=

∣

∣

∣

∣

Rnn (^0) nn
Snn Tnn

∣

∣

∣

∣

2 n

=|Rnn||Tnn|, (4.5.3)

A 2 n+1=

∣

∣

∣

∣

R

∗

n+1,n+1^0 n+1,n

S

∗

n,n+1 T

∗

nn

∣

∣

∣

∣

2 n+1

=|R

∗

n+1,n+1||T

∗

nn|. (4.5.4)

The method of factorization can be illustrated adequately by factorizing

the fifth-order determinantA 5 defined in (4.5.2).

A 5 =

∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣

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

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

c 1 c 2 c 3 c 2 c 1

b 5 b 4 b 3 b 2 b 1

a 5 a 4 a 3 a 2 a 1

∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣