I have computed the eigenvalues and eigenvectors of a matrix numerically.

However, the matrix is relatively small and sparse, and I would like to go back and compute them manually, to state them analytically (as exact equations).

Here is the [A] matrix:

$\displaystyle \begin{bmatrix} 0& 0& 0& 1\\ 1& 0& 0& 0\\ 0& 1& 0& 1\\ 0& 0& 1& 0\end{bmatrix}$

One of the eigenvalues is a REAL constant, k.

So, to solve for the associated eigenvector, the system becomes:

$\displaystyle \begin{bmatrix} -k& 0& 0& 1\\ 1& -k& 0& 0\\ 0& 1& -k& 1\\ 0& 0& 1& -k\end{bmatrix}$

which is set to zero.

I have reduced this system to the following:

$\displaystyle \begin{bmatrix} 1& -k& 0& 0\\ 0& 1& -k& 1\\ 0& 0& 1& -k\\ 0& 0& 0& 0 \end{bmatrix}$

So the eigenvector can be stated in terms of the fourth variable:

$\displaystyle \begin{bmatrix}-k(k^2+1)\\ (k^2+1)\\ k\\ 1\end{bmatrix}$

HOWEVER, this result does not confirm the result I got for this eigenvalue numerically. The numerical result is complex; it has both real and imaginary elements, and all elements are non-zero. The analytic result I get above is real only.

Anybody here willing to verify my work, and point out where I went wrong?