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?



Linearization of a vector field
Unfortunately, you need to be careful, as this is false as stated, unless you allow the chart to be only Lipschitz. Consider, for example, my answer to [this question](https://mathoverflow.net/questions/76971/nice-metrics-for-a-morse-gradient-field-counterexample-request), where a smooth example in dimension $2$ with $a\_1=1$ and $a\_2=2$ is given that cannot be linearized by a smooth (or even $C^2$) change of coordinates.


For a proof of the linearization result, you can look in any good book on dynamical systems, or consult the original paper of Hartman (Proc. Amer. Math. Soc. 11 (1960), 610-620).


