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?



Is there an easy way to tell if all eigenvalues of a unitary or self-adjoint matrix only have eigenvalues of multiplicity two?
Assuming it is feasible to compute the characteristic polynomial, $p(x)$, in your situation, (which can certainly be done in principle if you know the matrix entries) there is a simple strategy. Given the information you already have, all roots have multiplicity two if and only if ${\rm gcd}(p(x),p^{\prime}(x))$ has degree $n.$ Note that this does not require factorization of $p(x),$ it only requires the Euclidean algorithm for polynomials, which is simple to implement computationally (here, $p^{\prime}(x)$ just denotes the derivative of $p(x)$).


