Chemistry Reference
In-Depth Information
<
a
þ
b
1
Þ
1
=
2
Þ
1
=
2
l
¼
S
;
c
11
¼ð
2
þ
2S
;
c
21
¼ð
2
þ
2S
1
þ
ð
1
:
28
Þ
:
a
b
1
Þ
1
=
2
Þ
1
=
2
l
2
¼
S
;
c
12
¼ð
2
2S
;
c
22
¼ð
2
2S
under these assumptions, these are the elements of the square matrices
L
and C (Equations 1.21). Matrix multiplication shows that these matrices
satisfy the generalization of Equations (1.22):
CAC
CSC
CSC
¼ L;
¼
¼
1
ð
1
:
29
Þ
so that matrices A and S are simultaneously diagonalized under the
transformation with the orthogonal matrix C.
All previous results can be extended to square symmetric matrices of
order N, inwhich case the solution of the corresponding secular equations
must be found by numerical methods, unless use can bemade of symmetry
arguments.
1.2 PROPERTIES OF EIGENVALUES AND
EIGENVECTORS
It is of interest to stress some properties hidden in the eigenvalues
, (Equations 1.23), of the symmetric
c
1
c
2
ð
l
l
Þ
and eigenvectors
1
2
matrix A of order 2 with A
22
¼
A
11
¼ a
and A
21
¼
A
12
¼ b:
In fact, Equation (1.17) can be written:
2
ð
l
1
l
Þð
l
2
l
Þ¼
l
1
l
2
ð
l
1
þ
l
2
Þ
l
þ
l
¼
0
ð
1
:
30
Þ
so that:
A
12
2
¼ a
2
b
2
l
1
l
2
¼
A
11
A
22
¼
det A
ð
1
:
31
Þ
l
þ
l
¼
A
11
þ
A
22
¼
2
a ¼
tr A
ð
1
:
32
Þ
1
2
0
, the determinant of
matrix A, is expressible as the product of the two eigenvalues; the
coefficient of
In Equation (1.17), therefore, the coefficient of
l
, the trace of matrix A, is expressible as the sum of the
two eigenvalues.
From the eigenvectors of Equations (1.23) we can construct the two
square symmetric matrices of order 2:
l
