I am studying non-linear dynamical systems with the linearization method around an equilibrium point, but I don't get the geometrical meaning of complex eigenvalues. (Let's focus on a 2D case) For example, if I have a generic matrix like this: $ \mathbf{A} = \begin{bmatrix} -2 & 1 \\ 0 & -3 \\ \end{bmatrix} $ there two real eigenvalues $ \lambda_1 = -2 , \lambda_1 = -3 $, that means that eigenvectors are scaled by a factor equal to the respective eigenvalue.
Stability of the system occurs when both of the two eigenvalues are negative, if I have interpreted this correctly this means that all vectors belonging to the direction of an eigenvectors are scaled by a negative value, they are centered in the equilibrium point "trying to go away from it" but they are "pushed" to the equilibrium (thanks to negative real eigenvalues). Since we are in a linear (linearized) system vectors outside of the eigenvectors directions are affected "in the same way" (thanks to linearity property) so all the vectors exhibit a stable behaviour (always with the assumption of being near to the equilibrium point). How can I geometrically interpret complex eingenvalues (both conjugate and purely imaginary) ? Can someone explain it to me, possibly with also a numerical example. (My intuition is that they are a sort of rotation)