Physical Chemistry 3: — Chemical Kinetics — - Christian-Albrechts ...

Appendix D 283
I Eigenvectors x : For each eigenvalue , we can write an eigenvalue equation
In matrix notation, this becomes
Ax 1 = 1 x 1 (D.53)
Ax 2 = 2 x 2 (D.54)
. (D.55)
Ax = x (D.56)
with the diagonal eigenvalue matrix
Λ =
AX= X Λ
⎛
⎜
⎝
⎞
1 0 0
0 2 0
⎟
. . . ⎠
0 0
(D.57)
(D.58)
The different eigenvectors x are usually normalized to unity, i.e.,
|x| = ¡ x † x ¢ 12
=1
(D.59)
I Matrix diagonalization: The matrix of the eigenvectors X can be determined by
multiplying the equation
AX= X Λ
(D.60)
from the left with X −1 ,whichgives
X −1 AX= X −1 X Λ
(D.61)
i.e.,
y
X −1 AX= Λ
(D.62)
The determination of the eigenvector matrix is equivalent to finding a matrix X that
transforms the matrix A into diagonal form (Λ).
I
Example:
A =
µ
2 3
310
(D.63)
Eigenvalue equations:
Ax = x
(D.64)
or in matrix form:
AX= X Λ
(D.65)