Does Orthogonal matrix have complex eigenvectors with the same absolute value? If it is true, how can I prove it?

Question

Does Orthogonal matrix have complex eigenvectors with the same absolute value (or modulus or magnitude)? If it is true, how can I prove it?

Answer 1

Note that if $A$ has eigenvector $x$ associated with eigenvalue $\lambda$, then $kx$ is also an eigenvector for any non-zero $k \in \Bbb C$.

So, every matrix, orthogonal or otherwise, has a set of eigenvectors of identical length.

Answer 2

Let $A$ be an orthogonal matrix, $\lambda$ an eigenvalue of $A$ and $v$ a $\lambda$-eigenvector. Then $\|v\|^2=v\cdot v=(Av)\cdot (Av)=\|Av\|^2=|\lambda|^2\|v\|^2$ and hence $|\lambda|=1$.