I is well known that there is no explicit formula for the eigenvalues of a general matrix (see *e.g.* [Wikipedia](http://en.wikipedia.org/wiki/Eigenvalues_and_eigenvectors#Eigenvalues)). This result is a consequence of (1) Abel's theorem, stating that there is no explicit formula for the roots of a general polynomial of degree >=5, and (2) the fact that for every polynomial $p$ there is a matrix (the companion matrix) which has characteristic polynomial $p$.


My question: Is it true that there is no explicit formula for the eigenvalues of a general **Hermitian** matrix? Or in other words: is there a way to find, for a given polynomial $p$, a **Hermitian** matrix which has characteristic polynomial $p$?



the impossibility of exactly computing eigenvalues
[A real symmetric tridiagonal matrix with a given characteristic polynomial](http://www-users.york.ac.uk/~slow500/reprints/1993SchmeisserGivenCharactersticPolynomial.pdf), G. Schmeisser (1993)


earlier work (behind a paywall):


[Expressing a polynomial as the characteristics polynomial of a symmetric matrix](http://www.sciencedirect.com/science/article/pii/0024379590903235), M. Fiedler (1990)


