I am interested in a class of $2n\times 2n$ unitary matrices with complex entries (if you prefer, we can replace "unitary" with "self-adjoint").

I know that all the eigenvalues of matrices in this class have (algebraic) multiplicity one or two. Some very interesting phenomena happens when all the eigenvalues have multiplicity two. Is there a way for me to tell when that happens based on the entries of the matrix, other than computing them by brute force?