3 Intermediate Determinant Theory
3.1 Cyclic Dislocations and Generalizations...........
Define column vectorsCjandC
∗
j
as follows:
Cj=
[
a 1 ja 2 ja 3 j···anj
]T
C
∗
j=
[
a
∗
1 ja
∗
2 ja
∗
3 j···a
∗
nj
]T
where
a
∗
ij=
n
∑
r=1
(1−δir)λirarj,
that is, the elementa
∗
ij
inC
∗
j
is a linear combination of all the elements
inCjexceptaij, the coefficientsλirbeing independent ofjbut otherwise
arbitrary.
Theorem 3.1.
n
∑
j=1
∣
∣
C 1 C 2 ···C
∗
j···Cn
∣
∣
=0.
Proof.
∣
∣C
1 C 2 ···C
∗
j
···Cn
∣
∣=
n
∑
i=1
a
∗
ij
Aij
=
n
∑
i=1
Aij
n
∑
r=1
(1−δir)λirarj.