To find the eigenvalues, you need to evaluate . Expand along the top row to see that . Thus satisfies the difference equation , together with the initial conditions and .
The auxiliary equation has solutions , where and . Standard techniques for solving difference equations (as described here, for example) show that , where .
The eigenvalues are given by . So they are the solutions of , or . Since , it follows that . (You can then use the fact that to write the eigenvalues as .)
Once you know the eigenvalues, it should be straightforward to get the eigenvectors, by solving the equations .