Let $A$ be an $n\times n$ matrix and $p(x)$ a polynomial over $\mathbb{C}$. How is the algebraic and geometric multiplicity of some eigenvalue $\lambda$ of $A$ related to the respective multiplicities of $p(\lambda)$ (as an eigenvalue of $p(A)$)?
By reducing to the case where $A$ is in Jordan canonical form, I think I've shown that algebraic multiplicity is preserved. I also think I can show that geometric multiplicity of $p(\lambda)$ (as an eigenvalue of $p(A)$) is at most the geometric multiplicity of $\lambda$ (as an eigenvalue of $A$).
I wanted to first check that the above claims were correct, and if so, whether there's a sharper bound for the geometric multiplicity of $p(\lambda)$.
Thanks!