After substitutions and simplifications, we end up with , where is the eigenvector associated with the eigenvalue
To find the eigenvalues, we must find where
Now to your question
Your coefficient matrix is
Thus, to find the eigenvalues, they must satisfy the equation
Thus, we see that
We the case of repeated eigenvalues.
The eigenvector for would be found by solving the equation
You end up with the equations , and
Since , we see that
Thus, we can let the eigenvector be
Now, let's look at what happens when
We have the equation
We come up with the equations and
Letting , we end up with
We can have one of the eigenvectors being
However, I'm not able to see another possible eigenvector!!!
Up to this point, our solution is
Anyone can chip in, if they'd like...I have a feeling there isn't another eigenvector...
Does this make sense?