If $k$ is an eigenvalue of $A$ of algebraic multiplicity $r$, then is $p(k)$ an eigenvalue of $p(A)$ of algebraic multiplicity $r$?

Question

Let $k \in \mathbb C$ be an eigenvalue of $A \in M(n,\mathbb C)$ of algebraic multiplicity $r$ (i.e. $k$ is an $r$-fold root of the characteristic polynomial of $A$). Let $p(x)$ be a polynomial with complex coefficients. Then is it true that $p(k)$ is an eigenvalue of $p(A)$ of algebraic multiplicity $r$ ? (I know that $p(k)$ is an eigenvalue of $p(A)$, but I am not sure about the algebraic multiplicity.)

Answer 1

Of course not. Consider $A=\pmatrix{1\\ &-1},\ k=-1$ and $p(x)=x^2$.