Q:

Complex matrix eigenvalues

Consider a square matrix with complex entries (e.g. a symplectic one). Let this matrix have a pair of complex conjugate eigenvalues: does this imply that the corresponding eigenvectors are complex conjugate too?
PS: I know that's a very basic and general question, so suggestions to improve it are welcome too

A:

In general, no. Take, for instance,$$A=\begin{bmatrix}-1 & -1-i \\ 1-i & 1\end{bmatrix},$$whose eigenvalues are $\pm i$. The eigenvectors corresponding to the eigenvalue $i$ are the multiples of $(1,-1)$, whereas the eigenvectors corresponding to the eigenvalue $i$ are the multiples of $(1,i)$.

