We've been doing some examples of showing when two specific systems are conjugate in class. I came across this more general question and I'm having trouble finding such a conjugacy.
Suppose there are two systems $(\mathbb{R}^n, F)$ and $(\mathbb{R}^n, G)$ where $F(x) = Mx + b$ and $G(x) = Mx$ for some matrix $M$ without eigenvalue 1. Then, there is a conjugacy between these two systems, that is, we can find a homeomorphism $h: \mathbb{R}^n \rightarrow \mathbb{R}^n$ such that $h \circ F = G \circ h$.
Any help would be greatly appreciated. Thank you in advance!
