I am studying Generalized Eigenvectors. It seems that we can define them as $\mathbf{p}_i$ in this equation:
$ (\mathbf{A}-\lambda\mathbf{I})^{k}\mathbf{p}_i = \mathbf{0} $
in which $k$ is the algebraic multiplicity of $\lambda$ in $ |\mathbf{A}-\lambda\mathbf{I}|=0$. Also it can be defined as:
$ (\mathbf{A}-\lambda\mathbf{I})\mathbf{p}_i = \mathbf{p}_{i-1},~~i=1\ldots,~~\mathbf{p}_{0}=\mathbf{0} $
or at least this is what I have learned (if sth is wrong, please let me know).
Are these definitions equivalent? I mean $ (\mathbf{A}-\lambda\mathbf{I})^{k}\mathbf{p}_i = \mathbf{0} $ if and only if $ (\mathbf{A}-\lambda\mathbf{I})\mathbf{p}_i = \mathbf{p}_{i-1} $?
I can prove the first one, if the second one is true (by multiplying both sides in $ (\mathbf{A}-\lambda\mathbf{I})$ k times), but how can I prove the second one, given the first one?
Thanks