In a paper that I was reading, I stumbled across the following theorem:

Let $X$ be a vector field with $$X= > a^ix^i\partial_{x^i} + > \mathcal{O}(|x|^2),$$ where $x$ is some chart and $a^i>0$. Then there exist a chart $y$ such that $X$ is linear with respect to $y$, meaning $$X =a^iy^i\partial_{y^i}.$$

It was referenced as "Sternbergs linearization theorem" and it sounded like common knowledge in the paper, but till now I couldn't find a proof anywhere.

Does anyone know a proof or a reference to one?

Also though my intuition is that this theorem does hold, I don't really have an understanding how it is important that the $a^i$ are greater $0$. Why does this make a difference?