Let $\mathbb{K}$ be a field and $p \in \mathbb{K}[X]$ a polynomial.
If $A,B \in \mathbb{K}^{n \times n}$ are similar, $p(A)$ and $p(B)$ are also similar
my attempt:
$A=S^{-1}BS$
$S,T\in \mathbb{K}^{n \times n}$
$p(B) = \sum^{n}_{i=0} a_i B^i$
$p(A)= \sum^{n}_{i=0} a_i(S^{-1}BS)^i$
$p(A) = T^{-1}p(B)T$
$\sum^{n}_{i=0} a_i(S^{-1}BS)^i = T^{-1} (\sum^{n}_{i=0} a_i B^i )T$
Im not sure how to solve this or if Im even on the right track
Hint
This is a consequence of the identity
$$p(A) = p(S^{-1} B S) = S^{-1} p(B) S$$ which is easy to verify due to linearity and the equality
$$(S^{-1} B S)^n = \underbrace{(S^{-1} B S) \dots (S^{-1} B S)}_{n \text{ times}} = S^{-1} B^n S$$ valid for all $n \in \mathbb N$.