Page 31 - Determinants and Their Applications in Mathematical Physics

3

Intermediate Determinant Theory

3.1 Cyclic Dislocations and Generalizations

Define column vectors C j and C as follows:
∗
j

T
C j = a 1j a 2j a 3j ··· a nj
T
C = a ∗ a ∗ a ··· a ∗
∗
∗
j 1j 2j 3j nj
where
n

a = (1 − δ ir )λ ir a rj ,
∗
ij
r=1
∗
that is, the element a ∗ in C is a linear combination of all the elements
ij j
in C j except a ij , the coefficients λ ir being independent of j but otherwise
arbitrary.
Theorem 3.1.
n

C 1 C 2 ··· C ··· C n =0.
∗

j
j=1
Proof.
n

∗ ∗

C 1 C 2 ··· C ··· C n =
j a A ij
ij
i=1
n n

= A ij (1 − δ ir )λ ir a rj .
i=1 r=1