# How to determine the N-smallest eigenvalues of a symmetric matrix using the Power Method?

I was assigned to make a program that finds the largest, the N-largest, the smallest and the N-smallest eigenvalues of a symmetric matrix, using the Power Method. So far, I've been able to succesfully calculate the largest eigenvalue using the traditional Power Method, the N-largest using the Power Method with Deflation, and the smallest using the Inverse Iteration (the Inverse Iteration as described here in section 3-2: Iterative Methods).

But, right now I have no idea how to determine the N-smallest. I tried using the Inverse Iteration with Deflation to calculate the N-smallest but it is not working. When I calculate the second smallest (and so on...) I don't get the expected results, as if it's not possible to simply apply the Inverse Iteration with deflation. What am I missing? What should be the right way to calculate de N-smallest?

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I assume by "largest" and "smallest" you mean in absolute value. The $n$'th smallest eigenvalue of $A$ is $\pm \sqrt{t - \mu}$ where $\mu$ is the $n$'th largest eigenvalue of $t I - A^2$ if $t$ is large enough. See how this moves if you replace $A$ by $A-\epsilon I$ for small $\epsilon$ and you can tell whether it's $+$ or $-$.

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Yes, I mean in absolute value, thanks for your reply. I was totally unaware of that way of calculating the N-smallest eigenvalues of a matrix, and definitely will keep it in mind, next time I need to do it. Now, since my homework requires me to do it using the Power Method, I wonder if you know how to determine the N-smallest eigenvalues using the Power Method, I feel like it's easy and I'm missing something obvious but I can't see it. – Robst Sep 17 '12 at 1:39

Is your matrix positive-definite ? (all eigenvalues positives). If yes, let $M$ be the largest eigenvalue. Find the N-largest eigenvalues of $A-MId$ by power method w/ deflation to find the N smallest eigenvalues of $A$.

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