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Hello, I am considering two real invertible $3\times 3$ matrices $A$ and $B$ and a nonzero vector $v\in\mathbb{R}^3$ and i am wondering if the submonoid $E$ of the monoid $(A,B)$ genererated by $A$ and $B$ with $v$ as an eigenvector

$$E:= \left( C\in (A,B), \ v \text{ is an eigenvector of } C \right)$$

is finitely generated.

Does it even hold for $2\times 2$ matrices? Thank you in advance.

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