## Similar linear transformations have the same characteristic and minimal polynomials

Let $V$ be an $n$-dimensional vector space over a field $F$. Prove that similar linear transformations on $V$ have the same characteristic and the same minimal polynomial.

Suppose $A$ and $B$ are similar matrices (i.e. linear transformations) on $V$. Then we have an invertible matrix $P$ such that $B = P^{-1}AP$.

Now $\mathsf{char}(B) = \mathsf{det}(xI - B)$ $= \mathsf{det}(xI - P^{-1}AP)$ $= \mathsf{det}(P^{-1}xP - P^{-1}AP)$ $= \mathsf{det}(P^{-1})\mathsf{det}(xI - A)\mathsf{det}(P)$ $= \mathsf{det}(xI - P)$ $= \mathsf{char}(A)$. So $A$ and $B$ have the same characteristic polynomial.

Recall that the minimal polynomial of a transformation $T$ is the unique monomial generator of $\mathsf{Ann}(V)$ in $F[x]$ (under the usual action of $F[x]$ on $V$). Thus, to show that $A$ and $B$ have the same minimal polynomial, it suffices to show that they have the same annihilator in $F[x]$ under their respective actions on $V$.

To this end, suppose $p(x) \in \mathsf{Ann}_A(V)$ (where $\mathsf{Ann}_A$ denotes the annihilator induced by $A$). Then as a linear transformation, we have $p(A) = 0$. Say $p(x) = \sum r_ix^i$; then $\sum r_iA^i = 0$. But $A^i = PB^iP^{-1}$, so that $\sum r_iPB^iP^{-1} = 0$, and thus (left- and right-multiplying by $P^{-1}$ and $P$, respectively) $\sum r_iB^i = 0$. So $p(B) = 0$ as a linear transformation, and we have $p(x) \in \mathsf{Ann}_B(V)$. The reverse inclusion is similar, so that $\mathsf{Ann}_A(V) = \mathsf{Ann}_B(V)$ as desired.