I have a large numerical matrix (the weighted laplacian matrix of a weighted graph). I compute the eigenvalues and eigenvectors numerically (MATLAB), and when I look at the result I suspect that several of the eigenvalues are degenerate, even though they are numerically distinct (small numerical differences). Is there a way to test to see if they are really distinct eigenvalues or not?
For example, I had considered taking dot products of their corresponding eigenvectors, but there are a couple of issues with that: (1) distinct eigenvectors for an eigenvalue with geometric multiplicity > 1 are linearly independent, but not necessarily orthogonal (though they can be made so), and (2) even if they were orthogonal that wouldn't tell my anything since the subspaces corresponding to distinct eigenvalues are also orthogonal to each other. Any ideas?
Additional info: The matrices that I am working with are real symmetric matrices, and the numerical values vary by as much as 7 orders of magnitude (i.e. the edge weights of the graph span 7 orders of magnitude).