# Integer eigenvectors

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.

• Try linalg.org Aug 11 '17 at 0:33

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.