Finding eigenvectors for the largest eigenvalue vs one with the largest absolute value

If I want to solve a generalized eigenvalue problem such as:

$$A x = \lambda x$$

The problem is to find eigenvectors corresponding to the largest eigenvalues (sometimes in an optimization problem that form it as a generalized eigenvalue problem).

I notice some people solve this problem by finding eigenvectors corresponding to the largest eigenvalues in absolute value.

Are these two method similar? What are the difference between them? I mean in what situation I should use eigenvalues in absolute value?

• Could you provide some links with methods you say? I cannot help but I'm interested in methods providing eigenvectors corresponding to the maximum absolute value of eigenvalue. Thanks! Aug 23, 2014 at 4:01
• I don't have links actually. I am trying to solve the generalized eigenvalue problem and not sure which one should I use and what is the difference. Aug 23, 2014 at 4:14
• Thanks fir the response, @Matthew! Aug 23, 2014 at 4:38

If you need to find the eigenvector with largest eigenvalue, you must modify the procedure. Sometimes you know that the largest eigenvalue is also the largest-magnitude eigenvalue (for example, if your matrix is positive-definite) and so you can again use power iteration. Otherwise, you can find the largest-magnitude eigenvalue; if it is positive, you are done. Otherwise, construct a new spectrally-shifted matrix $M' = M - \lambda I$ where $\lambda$ is the largest-magnitude, negative, eigenvalue. The eigenvalues of $M'$ are now all non-negative, so find its largest-magnitude eigenvalue $\mu$ and corresponding eigenvector $v$; the largest eigenvalue of $M$ is then $\mu+\lambda$ with eigenvector $v$.