I'm currently studying the stability conditions for symplectic matrices, but I've some doubts that probably stem directly from basic linear algebra.
Let $M$ be a 4x4 real symplectic matrix.
Let $V$ be its eigenvector, with $e^{i\theta}$ as eigenvalue ($\theta \in \mathbb R$), i.e. $MV = e^{i \theta}V$. It is possibile to prove that in this case we have that $e^{-i \theta}$ is another eigenvalue with eigenvector $V^*$, i.e. $MV^* = e^{-i \theta}V^*$ (where the superscript $^*$ stands for "complex conjugate").
Suppose there is too a generalized eigenvector $W$, i.e. $MW = e^{i \theta}W + V$.
- Does it mean that there's too a complex conjugate generalized eigenvector $W^*$, i.e. $MW^* = e^{-i \theta}W^* + V^*$?
- How many invariant bidimensional subspaces does the matrix $M$ have?
- Depending on the eigenvalues ($e^{\pm i \theta} = \pm 1$ or $e^{\pm i \theta} \neq \pm 1$), is there a difference in the number of those invariant subspaces?