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