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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.

user602672
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  • First step: assume $A$ is upper triangular. Is it easier to prove there? How might your proof change if we throw in the $Q$s? – user3482749 Jan 04 '19 at 18:00

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Hint: exhibit an eigenvector $y(\neq 0)$ of $U$, and note that $x=Qy$ is then an eigenvector of $A$.

metamorphy
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