1. ## Help Computing Eigenvectors

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?

2. ## Re: Help Computing Eigenvectors

I haven't been through the arithmetic, but I can offer a few comments.
Complex eigenvalues occur as conjugate pairs, in which case there can't be a single real eigenvalue, there have to be none, two or four.
If the original matrix contains only real elements and an eigenvalue is real, then the associated eigenvector will contain real components only.
Check your algebra, I think that the $\displaystyle (k^{2}+1)$'s should be $\displaystyle (k^{2}-1)$'s, and I think that the negative sign at the front of the first component shouldn't be there.

3. ## Re: Help Computing Eigenvectors

Thanks, Bob.

You are correct; the signs were incorrect for two of the elements.

And I was mis-reading the values of the eigenvalues.
The results were output as part of a matrix and I was reading the second column as the column vector containing the imaginary components of the eigenvector. Instead, I should have been reading it as a completely different--solely real--eigenvector, the one corresponding to the other real eigenvalue.

It is a reminder how useful a second pair of eyes can be.

Once again, thank-you.

It all works now.