Consider the linear dynamical system

that is

We look for solutions on the form

If we instert these to the system we get

that is

This equation system has nontrivial solutions if and
only if

that is, if and only if

This is the *characteristic
equation* of the system and the solutions its *eigenvalues*.
If we now for each eigenvalues solve the equation system

we obtain the eigenvectors

Then we have the general solution of our original
dynamical system as

that is

where *C*_{1} and *C*_{2} are arbitrary constants.

Example 8: Assume that the matrix *A* is

This matrix has characteristic equation

with the eigenvalues

* *

and corresponding eigenvectors

This means that the general solution of the
corresponding dynamical system is

that is

**Example 9:** Solve the system

**Solution:** We consider the matrix

and its characteristic equation

that is

The eigenvalues thus are

with corresponding eigenvectors

This means that the dynamical system has the general
solution

that is

These are all *complex *solutions.
We are actually only interested in the *real* solutions. With
help of Euler's
formula we get

If we now pick arbitrary real constants *D*_{1} and *D*_{2}
and put

we get the general real solution

Example 8:

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