I am using a two different computational libraries to calculate the eigenvectors and eigenvalues of a symmetric matrix. The results show that the eigenvalues calculated with both libraries are exactly the same, however, the eigenvectors differ. Nevertheless, both seem to be correct since their eigenvectors are orthogonal and the factorization is also correct. How can that be possible? I asked the developers and although they seemed confused, they said that "probably" has to do with the directions of the vectors.
If it is possible, is there a kind of margin error between those two different range of eigenvectors so I could compare them and make sure they are in range (they are right)?