Question

When can a asymmetric matrix has complex eigenvalues?

Answer 1

A Hermitian matrix always has real eigenvalues and real or complex orthogonal eigenvectors. A real symmetric matrix is a special case of Hermitian matrices, so it too has orthogonal eigenvectors and real eigenvalues, but could it ever have complex eigenvectors?

My intuition is that the eigenvectors are always real, but I can't quite nail it down.