Non-symmetric matrix projection

I am performing minimization for a N=10 parameter function. If I have, say, 100 trial solution vectors, where each solution vector is a row in , my goal is to use the top 10 solutions to form a 10 x 10 matrix , perform eigenanalysis on it, and then get the eigenvectors to reflect which step direction the group of solutions is heading. Certainly, is not symmetric, so I would need to solve the general eigenvalue problem for a real non-symmetric matrix.

In order to find the single direction of the solutions, wouldn't I need to add together the eigenvectors for the 2 greatest eigenvalues? Basically, how can I obtain the single vector reflecting the single direction in which the top 10 solutions are heading?