Now; A
À1 A ¼
cos a sin a 0
À sin a cos a 0
0
0
1
0
B
@
1
C
A
cos a sin a 0
À sin a cos a 0
0
0
1
0
B
@
1
C
A
¼
cos
2
a þ sin
2
a
À cos a sin a þ cos a sin a 0
À sin a cos a þ sin a cos a
sin
2
a þ cos
2
a
0
0
0
1
0
B
@
1
C
A
¼
1 0 0
0 1 0
0 0 1
0
B
@
1
C
A ¼ E
Similarly; AA
À1
¼
cos a À sin a 0
sin a cos a 0
0
0
1
0
B
@
1
C
A
cos a sin a 0
À sin a cos a 0
0
0
1
0
B
@
1
C
A
¼
cos
2
a þ sin
2
a
cos a sin a À cos a sin a 0
sin a cos a À sin a cos a
sin
2
a þ cos
2
a
0
0
0
1
0
B
@
1
C
A
¼
1 0 0
0 1 0
0 0 1
0
B
@
1
C
A ¼ E
) A
À1 A ¼ AA
À1
¼ E
(iv) Orthogonal Matrix
An orthogonal matrix is a unitary matrix whose all elements are real. One can
immediately see that A
−1 = A
T if A is orthogonal.
Since AA
T = I, it follows that
X n
k¼0
A ik A jk ¼ d ij
¼ 1; if i ¼ j
¼ 0; if i 6 ¼ j
This can be stated as:
(i) the sum of the squares of the elements in any row (column) is equal to 1, that
is, A
2
11 þ A
2
12 þ A
2
13 ¼ 1, etc. and
(ii) the sum of the products of elements from one row (column) and the corresponding element of another row (column) is equal to zero, that is,
A 11 A 21 + A 12 A 22 + A 13 A 23 = 0, etc.
236
6 Unit Cell Symmeteries and Their Representations