4.3 Skew-Symmetric Determinants 67
[
A
(2n−1)
ij
] 2
=A
(2n−1)
ii A
(2n−1)
jj. (4.3.9)
It follows from the section on bordered determinants (Section 3.7.1) that
∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣
x 1
A 2 n− 1
.
.
.
......... x 2 n− 1
y 1 ···y 2 n− 1 •
∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣
2 n
=−
2 n− 1
∑
i=1
2 n− 1
∑
j=1
A
(2n−1)
ij
xiyj. (4.3.10)
Putxi=ai, 2 nandyj=−aj, 2 n. Then, the identity becomes
A 2 n=
2 n− 1
∑
i=1
2 n− 1
∑
j=1
A
(2n−1)
ij
ai, 2 naj, 2 n (4.3.11)
=
2 n− 1
∑
i=1
2 n− 1
∑
j=1
[
A
(2n−1)
ii
A
(2n−1)
jj
] 1 / 2
ai, 2 naj, 2 n
=
[
2 n− 1
∑
i=1
[
A
(2n−1)
ii
] 1 / 2
ai, 2 n
]
2 n− 1
∑
j=1
[
A
(2n−1)
jj
] 1 / 2
aj, 2 n
=
[
2 n− 1
∑
i=1
[
A
(2n−1)
ii
] 1 / 2
ai, 2 n
] 2
. (4.3.12)
However, eachA
(2n−1)
ii
,1≤i≤(2n−1), is a skew-symmetric determinant
of even order (2n−2). Hence, if each of these determinants is the square
of a polynomial function of its elements, thenA 2 nis also the square of a
polynomial function of its elements. But, from (4.3.7), it is known thatA 2
is the square of a polynomial function of its elements. The theorem follows
by induction.
This proves the theorem, but it is clear that the above analysis does not
yield a unique formula for the polynomial since not only is each square root
in the series in (4.3.12) ambiguous in sign but each square root in the series
for eachA
(2n−1)
ii
,1≤i≤(2n−1), is ambiguous in sign.
A unique polynomial forA
1 / 2
2 n
, known as a Pfaffian, is defined in a later
section. The present section ends with a few theorems and the next section
is devoted to the solution of a number of preparatory lemmas.
Theorem 4.11. If
aji=−aij,
then
a.|aij+x| 2 n=|aij| 2 n,
b.|aij+x| 2 n− 1 =x×
(
the square of a polyomial function
of the elementsaij