Determinants and Their Applications in Mathematical Physics

3.6 The Jacobi Identity and Variants 41

=

∣ ∣ ∣ ∣ ∣ ∣ ∣

Aa 21 a 23

Aa 31 a 33

a 11 a 13

a 41 a 43

∣ ∣ ∣ ∣ ∣ ∣ ∣

=A

2

∣

∣

∣

∣

a 11 a 13

a 41 a 43

∣

∣

∣

∣

=A

2

M 23 , 24

=σA

2

A 23 , 24.

Hence, transposingJ,

J=

∣

∣

∣

∣

A 22 A 24

A 32 A 34

∣

∣

∣

∣

=AA 23 , 24

which completes the illustration.

Restoring the parametern, the Jacobi identity withr =2,3 can be

expressed as follows:

r=2:

∣

∣

∣

∣

∣

A

(n)

ip

A

(n)

iq

A

(n)

jp

A

(n)

jq

∣

∣

∣

∣

∣

=AnA

(n)

ij,pq. (3.6.4)

r=3:

∣ ∣ ∣ ∣ ∣ ∣ ∣

A

(n)

ip A

(n)

iq A

(n)

ir

A

(n)

jp

A

(n)

jq

A

(n)

jr

A

(n)

kp

A

(n)

kq

A

(n)

kr

∣ ∣ ∣ ∣ ∣ ∣ ∣

=A

2

n

A

(n)

ijk,pqr

. (3.6.5)

3.6.2 The Jacobi Identity — 2

The Jacobi identity for small values ofrcan be proved neatly by a technique

involving partial derivatives with respect to the elements ofA. The general

result can then be proved by induction.

Theorem 3.4. For an arbitrary determinantAnof ordern,

∣

∣

∣

∣

A

ij

n

A

iq

n

A

pj

n

A

pq

n

∣

∣

∣

∣

=A

ip,jq

n

,

where the cofactors are scaled.

Proof. The technique is to evaluate∂A

ij

/∂apqby two different methods

and to equate the results. From (3.2.15),

∂A

ij

∂apq

=

1

A

2

[

AAip,jq−AijApq

]

. (3.6.6)

Applying double-sum identity (B) in Section 3.4,

∂A

ij

∂apq

=−

∑

r

∑

s

∂ars

∂apq

A

is

A

rj