Chemistry Reference
In-Depth Information
Combining (4.41) and (4.37) leads to a single equation for the vector
u
, namely
A C
.
D
I
/
C R
1
u
D 0
:
1
ˇ
S
b
1
T
(4.42)
If
u
D 0
, then from (4.41),
v
D 0
, and from (4.40),
w
D
0. Since the eigenvector has
to be nonzero,
u
must differ from zero, so the determinant of the coefficient matrix
has to be zero.
We will next rewrite this matrix in the special form (E.2) introduced in
Appendix E, so the characteristic polynomial of the system can be obtained in a
simple form. A straightforward calculation shows that the coefficient matrix can
take the form
1
ˇ
S
.
D
I
/
b
1
T
;
a
1
T
.
I D
/
C
.
D
I
/
R
1
where
a D
.a
1
;:::;a
N
/
T
. The determinant of this matrix can be factored as
det
.
D
I
/
R
1
.
I D
/
det
I C
..
D
I
/
R
1
.
I D
//
1
1
T
D
0:
1
ˇ
S
a
.
D
I
/
b
(4.43)
The first factor of the last equation is zero if
1
a
k
r
k
a
k
D
0;
which implies that
D
1
a
k
.1
C
r
k
/:
(4.44)
The second factor can be simplified by using identity (E.1), and the resulting
equation is
.
I D
//
1
1
ˇ
S
.
D
I
/
b
1
C
1
T
..
D
I
/
R
1
a
D
0;
that is,
ˇ
S
.1
a
k
/
ˇ
S
1
a
k
r
k
a
k
X
N
1
C
D
0;
a
k
kD1
which can be rewritten as
N
X
r
k
Œ.a
k
C
ˇ
S
/
ˇ
S
1
a
k
.1
C
r
k
/
D
ˇ
S
:
(4.45)
k
D
1
Search WWH ::
Custom Search