Eigenvectors of $A \in SO(2n)$ and $A \in SO(2n+1)$

Question

Does every matrix $A \in SO(2n)$ have an eigenvector?

Does every matrix $A \in SO(2n+1)$ have an eigenvector?

I think that you can answer both questions with yes, is that true?

Answer 1

Every matrix has eigenvectors over the complex numbers. If the matrix is real, there are real eigenvectors for every real eigenvalue, but none for a non-real eigenvalue. Assuming you want real eigenvectors, the questions become, does every matrix in $SO(2n)$ or $SO(2n+1)$ have at least one real eigenvalue?

Answer 2

For the first question you just need to write down a rotation in $SO(2)$ which does not send any non-zero vector $v$ to $\pm v$. In fact it is harder to not do this than to do it.

For the second question, note that there will be an odd number of eigenvalues and they are closed under complex conjugation. What does this tell you that at least one of the eigenvalues must be?