Consider a two-by-two matrix $A$ which has real entries with purely imaginary eigenvalues. I am wondering if one can deduce that $A$ under some basis is given by
$$ \begin{pmatrix}0 & \lambda \\ -\lambda &0 \end{pmatrix}. $$
My thoughts: I think this is probably false thinking about this as some sort of Jordan block. However, I am finding it very difficult to find a countable example as one needs to show that there doesn't exist a change of basis.
Context: This arises out of me thinking about dymanical systems. More precisely, I have seen people use this jump when thinking about centers, however, I am not sure if this is generally true.