I is well known that there is no explicit formula for the eigenvalues of a general matrix (see *e.g.* Wikipedia). 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 $p$?