Chemistry Reference
In-Depth Information
therefore becomes:
C
AC ¼
L
ð
2
36
Þ
:
Namely, a Hermitian matrix
A
can be brought to diagonal form by a
unitary transformation with the complete matrix of its eigenvectors.
We examine below the simple case of the 2
2 Hermitian matrix
A
:
1 S
S 1
A ¼
ð
2
37
Þ
:
The secular equation is
¼
0
1
l
S
ð
2
38
Þ
:
S
1
l
giving upon expansion the quadratic equation in
l
:
2
l
þ
1
S
2
l
2
¼
0
ð
2
39
Þ
:
with the roots (the eigenvalues):
l
¼
1
S
;
l
¼
1
þ
S
ð
2
40
Þ
:
1
2
We now turn to the evaluation of the eigenvectors.
(i)
l
1
¼
1
S
(
ð
1
l
1
Þ
c
1
þ
Sc
2
¼
0
c
1
þ
c
2
¼
1
ð
2
41
Þ
:
We solve the homogeneous linear system (2.28) for the first
eigenvalue with the additional constraint of coefficients normal-
ization:
1
1
¼
l
c
2
c
1
1
S
¼
S
S
¼
1
)
c
2
¼
c
1
1
ð
2
42
Þ
:
1
Solution of the homogeneous system is seen to give only the ratio c
2
/c
1
.
