I am doing a linear stability analysis of a 6-dimensional system, what I want to know is if the system is stable at numerically solved steady states by looking at the eigenvalues of the jacobian matrix. I sampled lots of parameters for the system, and noticed that most of the time I can get two eigenvalues with positive real parts, and the rest of the eigenvalues all have negative real parts. Looking back at the Jacobian, I noticed that they all conform to the following pattern $$\left( \begin{matrix} -x & -x & -x & 0 & u_1 & -v_1 \\ -y & -y & -y & -v_2 & 0 & u_2 \\ -z & -z & -z & u_3 & -v_3 & 0 \\ 1 & 0 & 0 & -1 & 0 & 0 \\ 0 & 1 & 0&0&-1&0 \\ 0&0&1&0&0&-1 \end{matrix}\right) $$ All the letters in the matrix represent positive number in $(0, 5)$. In the end I tried to sample directly this kind of matrices, and found that if I sample uniformly its nonzero entries, it is very rare (19/10000) that all its eigenvalues have negative real parts.
So my question is this, given this particular form of jacobian matrix, what can I infer about its eigenvalues (both deterministically and statistically, other than just by doing brute force sampling?) I also wonder if there is a mathematics branch that deals with similar problems.
Any hints about where I should look is appreciated. Thanks in advance!