Answer by Easy for Eigenvalues and Eigenvectors of Large Matrix

I think there is generally no easy method to find the eigenvalues as some of the matrices might even not have nice eigenvalues, like complex numbers in your case. For dimension higher than 5, it is well known that a polynomial of degree 5 could have no explicit solution. But sometimes one might be able to find its eigenvectors and hence find its eigenvalues, however this only applies to some cases.