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Is there a known way or software to find integer eigenvectors for an integer matrix with integer eigenvalues?

In particular, I have a large real symmetric matrix with only a small number of distinct eigenvalues. I want to know if it is possible to find an eigenbasis such that each eigenvector contains only -1, 0, and 1 entries.

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For an integer matrix $A$ and an integer eigenvalue $\lambda$, $A - \lambda I$ is an integer matrix, and Gaussian elimination will produce rational eigenvectors; multiply by a common denominator and you'll get an integer eigenvector.

But the question in your second paragraph is different: there is an eigenvector with entries $-1, 0, 1$ iff there are two different sets of columns of $A - \lambda I$ with the same sum. Although it's not guaranteed to work for this, I might try to find these using the LLL algorithm.

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