I am trying to understand the intuition behind eigenvalues/eigenvectors through the lens of repeated matrix multiplication:
Given a $2\times2$ matrix $M$ and $2D$ vector $v$, multiplying $v$ repeatedly with $M$ causes the result ($M^n v$) to gravitate towards one of the eigenspaces of $M$ because:
$$M^n v = M^n(\alpha x_1 + \beta x_2) = (\alpha \lambda_1^n x_1 + \beta \lambda_2^n x_2)$$
where $x_1$ and $x_2$ are eigenvectors of $M$ and $\lambda_1$ and $\lambda_2$ the corresponding eigenvalues. As $n$ gets larger $M^n v$ will gravitate towards either $\alpha \lambda_1^n x_1$ or $\beta \lambda_2^n x_2$, whichever has the dominant eigenvalue.
assuming: $v = \alpha x_1 + \beta x_2$
So the above is a way to connect the abstract concept of eigenvalue/eigenvector to something concrete: what happens when you apply a matrix over and over to a vector.
However, the intuition breaks down for me with complex eigenvectors. I know repeated multiplication by a matrix with complex eigenvectors causes the result to either spiral outwards or inwards.
Is there simple math such as above to see why?
Edit: I know similar questions have been asked before, but I ask in the context of repeated matrix multiplication