Chemistry Reference
In-Depth Information
Equation (2.29) is known as characteristic (or secular) equation of the
square matrix
A
. It is an algebraic equation of degree n in
l
, with
l
;
...
;
l
n
n roots
ð
the eigenvalues of
AÞ
c
1
;
c
2
;
...
;
c
n
;
l
1
2
ð
2
30
Þ
:
n column coefficients
ð
the eigenvectors of
AÞ
The whole set of the n eigenvalue equations for
A
Ac
1
¼
l
1
c
1
;
Ac
2
¼
l
2
c
2
;
...
;
Ac
n
¼
l
n
c
n
ð
2
31
Þ
:
can be replaced by the full eigenvalue equation
AC ¼ C
L
ð
2
32
Þ
:
if we introduce the following square matrices of order n:
0
@
1
A
l
1
0
0
0
0
00
l
n
l
2
L
¼
;
0
@
1
A
c
11
c
12
c
1n
c
21
c
22
c
2n
C ¼ðc
1
c
2
c
n
Þ¼
ð
2
33
Þ
:
c
n1
c
n2
c
nn
where
is the diagonal matrix of the n eigenvalues and
C
is the rowmatrix
of the n eigenvectors, a square matrix on the whole.
If det
C
=
L
0, then
C
1
exists and the square matrix
A
can be brought to
diagonal form through the transformation
C
1
AC ¼
L
ð
2
34
Þ
:
a process which is called the diagonalization of matrix
A
.
If
A
is Hermitian
A ¼ A
ð
2
35
Þ
:
then eigenvalues are real and eigenvectors orthogonal, so that the com-
pletematrix of the eigenvectors is a unitarymatrix (
C
1
¼ C
), and (2.34)
