$\newcommand{\R}{\mathbb{R}}$ Given two $2\times2$ matrices $A,B$ such that $h(e^{tB}(X)) = e^{tA}(h(X))$ where $h$ is a diffeomorphism. I want to prove that $A$ and $B$ are conjugated matrices.
I think I must differentiate both sides to get something resembling a conjugation. However, I am shamefully having troubles differentiating:
I do the following:
I can think of the maps involved as $h:\R^2\rightarrow\R^2$ and $e^{tA}:\R^2\rightarrow\R^2$. Let $p\in \R^2$. The following must be equal:
$$ D_p(h\circ e^{tB}) = D_{e^{tB}(p)}h\circ D_pe^{tB} \\ D_p( e^{tA}\circ h) = D_{h(p)}e^{tA}\circ D_ph $$
But now I do not know how to procced to differentiate this.