If a matrix $A$ is complex orthogonally similar to an upper triangular matrix, that is, $A=QUQ^T, Q^TQ=I$ and $U$ is upper triangular matrix, then there exist at least one eigenvector $x$ of $A$ such that $x^Tx\neq 0.$
This is an exercise in Horn and Johnson. Don't know how to start. Any help or hint will be appreciated.
Hint: exhibit an eigenvector $y(\neq 0)$ of $U$, and note that $x=Qy$ is then an eigenvector of $A$.