If eigenvectors of a matrix are orthogonal, does that imply anything about the matrix (normal, hermitian, etc)? Normally, content related to my question proves the converse. For example, if a matrix is hermitian, then its eigenvectors corresponding to different eigenvalues are orthogonal.
If the eigenvectors of a square matrix under (separately) both $\mathbb{R}$ and $\mathbb{C}$ are orthonormal (or unitary), however, can we infer something about the matrix?

Complex case: If $M[e_1 ... e_n] = [e_1 ... e_n] \mbox{ diag } (\lambda_1 ... \lambda_n)$, or $MP = P\Lambda$ then $M=P\Lambda P^{-1} = P\Lambda P^*$. This is Hermetian iff $\Lambda$ is real (i.e. only real eigenvalues). In the real case if you assume that there are $n$ real eigenvectors then implicitly you have that all eigenvalues are real so the same argument shows that  $M$ is automatically symmetric in this case.
