Determinants and Their Applications in Mathematical Physics

3.3 The Laplace Expansion 25

is applied in Section 3.6.2 on the Jacobi identity. Formulas (3.2.16) and

(3.2.17) are applied in Section 5.4.1 on the Matsuno determinant.

3.3 The Laplace Expansion

3.3.1 A Grassmann Proof..................

The following analysis applies Grassmann algebra and is similar in nature

to that applied in the definition of a determinant.

Letisandjs,1≤s≤r,r≤n, denoterintegers such that

1 ≤i 1 <i 2 <···<ir≤n,

1 ≤j 1 <j 2 <···<jr≤n

and let

xi=

n

∑

k=1

aijek, 1 ≤i≤n,

yi=

r

∑

t=1

aijtejt, 1 ≤i≤n,

zi=xi−yi.

Then, any vector product is which the number ofy’s is greater thanror

the number ofz’s is greater than (n−r) is zero.

Hence,

x 1 ···xn=(y 1 +z 1 )(y 2 +z 2 )···(yn+zn)

=

∑

i 1 ...ir

z 1 ···yi 1 ···yi 2 ···yir···zn, (3.3.1)

where the vector product on the right is obtained from (z 1 ···zn) by replac-

ingzisbyyis,1≤s≤r, and the sum extends over all

(

n

r

)

combinations of

the numbers 1, 2 ,...,ntakenrat a time. They’s in the vector product can

be separated from thez’s by making a suitable sequence of interchanges

and applying Identity (ii). The result is

z 1 ···yi

1

···yi

2

···yi

r

···zn=(−1)

p

(

yi

1

···yi

r

)(

z 1

∗

···zn

)

, (3.3.2)

where

p=

n

∑

s=1

is−

1

2

r(r+ 1) (3.3.3)

and the symbol∗denotes that those vectors with suffixesi 1 ,i 2 ,...,irare

omitted.