when I'm learning markov matrix
when proving THEOREM 4.12, It said, characteristic polynomial of a matrix A $ch(A) =(x−c_1)^{a_1} ···(x−c_t)^{a_t} ⇒ ch(A^k) =(x−c_1^k)^{a_1} ···(x−c_t^k)^{a_t}$. why is that? I can understand that if $c_1$ is eigenvalue of A, then $c_k$ is eigenvalue of $A^k$,(and the eigenvector can be same), but why is the algebra multiplicity the same?
besides, the eigenvector of A and $A^k$ is same, but is it true for generalized eigenvector? Thanks.
If $A$ is a matrix in $\mathcal M_n(\Bbb C)$ then it's trigonalizable over $\Bbb C$ i.e. there's an invertible matrix $P$ such that
$$A=PTP^{-1}$$ where
$$T=\begin{pmatrix}c_1&\times&\cdots&\cdots&\times\\ &\ddots&&&\vdots\\ &&\ddots&&\times\\&&&&c_{a_t} \end{pmatrix}$$ is a triangular matrix and $c_i$'s their eigenvalues. Now we see that
$$A=PT^kP^{-1}$$ where $$T^k=\begin{pmatrix}c_1^k&\times&\cdots&\cdots&\times\\ &\ddots&&&\vdots\\ &&\ddots&&\times\\&&&&c_{a_t}^k \end{pmatrix}$$ so the characteristic polynomial of $T^k$ is
$$\chi_{T^k}(x)=\det(xI_n-T^k)=\prod_{i=1}^t(x-c_i^k)^{a_i}$$