Key Problem : Is there any theorem about eigenvalues or positive semi-definiteness of small size matrices with small integer elements?

I have to check positive semi-definiteness of many symmetric matrices with integer elements. First I used eigenvalues, but floating point round error happens : eig(in numpy) sometimes gives small negative eigenvalues when a matrix is actually positive semi-definite.

I know Sylvester's criterion for positive semi-definiteness can avoid this problem. But I really don't want to use it since it requires computation of determinant of all principal minors, and I have to deal with really many matrices($ > 10^{20}$). I have to do anything to reduce number of calculations.

All the elements are integer and have small absolute values($\ <3 $), sizes of matrices are also small($\ < 10 \times 10 $). It seems pretty good condition, so I believe someone already researched this kind of matrices. Does anyone know useful theorems for this situation?

P.S: I'm pretty newbie in English Internet community and not native English speaker. So if you find something awkward, pardon me and let me know.