Here are a few observations that may be useful.

First, The for all n.

This gives that so zero is an eigenvalue of multiplicity n-2.

When you compute and reduce to row echelon form you can see a pattern for the eigenvectors. Just write out a few small cases and look for the pattern.

For the case when is even we get

This implies that the polynomial is in the annihilating Ideal, so the minimum polynomial must divide

so the only possible eigenvalues are 0 and . The two cases I checked admitted 2 eigenvectors for the eigenvalue

When I computed the the matrix and reduced to row echelon form for n=4 and n=6 there seems to be a pattern for the eigenvectors . I hope this helps.

I haven't discovered anything useful for the case when n is odd.