Let $C$ be a matrix in companion form. Let us form a subset of general linear map $GL_n(\mathbb R)$ by \begin{align*} \mathcal E = \{G \in GL_n(\mathbb R): G = (g_1, Cg_1, \dots, Cg_1^{n-1})\}, \end{align*} where the first column $g_1$ of $G$ is a cyclic vector of $G$, i.e., $\{g_1, Cg_1, \dots, C^{n-1} g_1\}$ is a basis for $\mathbb R^n$. Obviously, $\lambda I \in \mathcal E$ for any $\lambda \mathbb R^{\times}$.
Can we classify this set? In particular, what are the topological properties we can say (open or closed, connected, etc) in the Euclidean topology?
