For a polynomial $p$, $Av=\lambda v$ implies $p(A)v=p(\lambda)v$. This follows by applying the power rule term by term. Over $\mathbb C$, the eigenvalues of $p(A)$ are exactly the values $p(\lambda)$ as $\lambda$ ranges over eigenvalues of $A$ (spectral mapping).
Like how this is organised?
Build your own knowledge tree on knowen.md: notes, decisions and tasks in one connected markdown graph, shared with your team and your AI agents.