Environmental Engineering Reference
In-Depth Information
For a square matrix with
N
rows and columns there are
N
eigenvalues, of which
some have to be counted several times. How many times an eigenvalue has to be
counted is not arbitrary, of course: we call it the order of the eigenvalue. In the
following example
there are the eigenvalues '1' and '2'. '1' has the order 2 and appears twice in the
MATLAB
output list. The unit matrix has only one eigenvalue ('1'), but with the
order of
N
. Try the command:
®
The nice property that the number of eigenvalues, if counted according to their
order, is equal to
N
is only true, if complex valued eigenvalues are allowed. In
general, eigenvalues are complex numbers, which is no problem for MATLAB
,of
®
course:
l
1
;
2
¼
2
i
, where
i
denotes the imaginary
The eigenvalues in this example are
1
p
(for complex numbers see also Chap. 15). Also, the eigenvectors can
be obtained from the
eig
command; see the MATLAB
®
help system for details.
The reader interested in the numerics of the calculation of eigenvectors should
consult a textbook on linear algebra.
The connection with linear systems of differential equations is that the eigenvalues
tell something about the behavior of the solutions. Negative real parts tell that
the unsteady solution is converging towards the equilibrium (which is given
by:
c
eq
¼
unit
i ¼
C
1
f
i
, see the preceding subchapter). The equilibrium in that case becomes
a steady state, as demonstrated in Fig.
18.7
. As both eigenvectors are negative in that
example, the solutions with increasing time tend towards the steady state.
The importance of eigenvalues for the analysis of the development of linear
systems is most apparent in
phase diagrams
. Phase diagrams, such as plots of the
phase space
, are a tool for the visualization of the behavior of linear and nonlinear
systems. In a phase diagram two dynamic variables or derivations are plotted
against each other. The representation is unique if there are only two unknown
variables. For the matrix
C
, treated in a listing above, the procedure is performed
and demonstrated by the following commands:
