Let $\mathcal E \subset M_n$ denote the set of matrices such that $e_1$ (standard basis vector) is a cyclic vector for all $A \in \mathcal E$. That is if $A \in \mathcal E$, $\{e_1, Ae_1, \dots, A^{n-1}e_1\}$ forms a basis for $\mathbb R^n$. Let us define a map $f: \mathcal E \to M_n(\mathbb R)$ by \begin{align*} A \mapsto B^{-1}AB, \end{align*} where $B = [e_1, Ae_1, \dots, A^{n-1}e_1]$. Now it is clear the image of $f$ will be a subset of matrices in companion form.
if we let $g : \mathcal E \to f(\mathcal E)$, my goal is to see:
Whether $g$ is injective. I was thinking in the wrong way in terms of kernel of $g$ but as pointed out in the comment, $g$ is not linear. But is $g$ injective?
If $g$ is injective, we should be able to find an inverse. Is it possible to give an explicit fourmula?