# The generalized eigenvector of $A$ matrix $A^k$ related to $A$

In these pdf,

It said,for characteristic polynomial of a matrix $A$ $ch(A) =(x−c_1)^{a_1} ···(x−c_t)^{a_t} ⇒ ch(A^k) =(x−c_1^k)^{a_1} ···(x−c_t^k)^{a_t}$. The corresponding algebra multiplicity is same, so what about geometric multiplicity? Do them also have relationship? And what about the corresponding generalized eigenvector? Do them also have a similar relationship? (For eigenvector, of course $A^k$ is same with $A$).
(I have proved corresponding generalized eigenvector of $A^k$ seem have no relationship with A, because $Av_1=\lambda A$, so obviously $A^2v_1=\lambda^2v_1$,$v_1$ is both eigenvector of A and $A^2$, but for degree 2 generalized eigenvector,suppose $v_2$ is generalized eigenvector of A, $Av_2=v_1+\lambda v_2$, then $A^2v_2=Av_1+\lambda Av_2$,then $A^2v_2=\lambda v_1+\lambda(v_1+\lambda v_2)=2\lambda v_1+\lambda^2v_2)$, no obvious relationship.)