Page 82 - Determinants and Their Applications in Mathematical Physics

4.3 Skew-Symmetric Determinants 67
2

(2n−1) (2n−1) (2n−1)
A = A A . (4.3.9)
ij ii jj
It follows from the section on bordered determinants (Section 3.7.1) that

x 1
. 2n−1 2n−1
. . (2n−1)
A 2n−1
= − A x i y j . (4.3.10)
......... x 2n−1 i=1 j=1 ij

y 1 ··· y 2n−1 •
2n
Put x i = a i,2n and y j = −a j,2n . Then, the identity becomes
2n−1 2n−1
(2n−1)
A 2n = A (4.3.11)
ij a i,2n a j,2n
i=1 j=1
2n−1 2n−1
(2n−1) (2n−1) 1/2

= A A
ii jj a i,2n a j,2n
i=1 j=1
 
2n−1 2n−1

(2n−1) 1/2 (2n−1) 1/2

= A  A 
ii a i,2n jj a j,2n
i=1 j=1
2
2n−1

(2n−1) 1/2

= A . (4.3.12)
ii a i,2n
i=1
(2n−1)
However, each A ,1 ≤ i ≤ (2n − 1), is a skew-symmetric determinant
ii
of even order (2n − 2). Hence, if each of these determinants is the square
of a polynomial function of its elements, then A 2n is also the square of a
polynomial function of its elements. But, from (4.3.7), it is known that A 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
(2n−1)
for each A ,1 ≤ i ≤ (2n − 1), is ambiguous in sign.
ii
1/2
A unique polynomial for A , known as a Pfaffian, is defined in a later
2n
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
a ji = −a ij ,
then
a. |a ij + x| 2n = |a ij | 2n ,

the square of a polyomial function
b. |a ij + x| 2n−1 = x × of the elements a ij