Page 101 - Determinants and Their Applications in Mathematical Physics

86 4. Particular Determinants

most general centrosymmetric determinant of order 5 is of the form

a 1 a 2 a 3 a 4 a 5

b 1 b 2 b 3 b 4 b 5

A 5 = c 1 c 2 c 3 c 2 c 1 . (4.5.2)

b 5 b 4 b 3 b 2 b 1

a 5 a 4 a 3 a 2 a 1
Theorem. Every centrosymmetric determinant can be factorized into two
determinants of lower order. A 2n has factors each of order n, whereas
A 2n+1 has factors of orders n and n +1.
Proof. In the row vector

R i + R n+1−i = (a i1 + a in )(a i2 + a i,n−1 ) ··· (a i,n−1 + a i2 )(a in + a i1 ) ,
the (n +1 − j)th element is identical to the jth element. This suggests
performing the row and column operations

1

R = R i + R n+1−i , 1 ≤ i ≤ n ,
i 2

C = C j − C n+1−j , 1 (n +1) +1 ≤ j ≤ n,
j 2
1 1
where n is the integer part of n. The result of these operations is
2 2
that an array of zero elements appears in the top right-hand corner of A n ,
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 notation M rs , etc., for a matrix with r rows and s columns. 0 rs is an
array of zero elements.

A 2n = R nn 0 nn

S nn
T nn
2n
= |R nn ||T nn |, (4.5.3)

R ∗
n+1,n+1
A 2n+1 = 0 n+1,n
S ∗ T ∗
n,n+1 nn 2n+1

= |R ∗ ||T ∗ |. (4.5.4)
n+1,n+1 nn

The method of factorization can be illustrated adequately by factorizing
the fifth-order determinant A 5 defined in (4.5.2).

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

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

c 1 c 2 c 3 c 2 c 1

A 5 =

b 5 b 4 b 3 b 2 b 1

a 5 a 4 a 3 a 2 a 1