Checking positive semi-definiteness of integer matrix 
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.
 A: You can tridiagonalize an integer matrix into an integer tridiagonal matrix using Householder reflections times integers.  The resulting tridiagonal matrix will be SPD iff the original is.  Sylvester’s criterion can be checked in linear time for tridiagonal matrices, since the determinants follow a recurrence relation:
If we are checking for positive definiteness, Sylvester's criterion can be evaluated in linear time via the recurrence since we only need to check $n$ principle minors.  For positive semidefiniteness, we must additionally check all $O(n^2)$ minors given by intervals of rows/columns.  This takes $O(n^2)$ time, which is still less than the tridiagonalization step.


*

*Tridiagonalization: https://math.byu.edu/~schow/resources/householder.pdf

*Recurrence: https://en.m.wikipedia.org/wiki/Tridiagonal_matrix
A: For small symmetric matrices, you could look at the characteristic polynomial.
The real symmetric matrix $A$ is positive semidefinite iff the coefficients of the characteristic polynomial are alternating in sign.  For $n \times n$ matrices this gives you $n$ integer expressions to check.
