Determinants and Their Applications in Mathematical Physics

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

)