$C^1$ eigenvectors of a Jacobian Matrix.

Question

Let $F: \mathbb{R}^2 \longrightarrow \mathbb{R}^2$ be a $C^2$ function given by $F(u,v)=(F_1(u,v),F_2(u,v))$. Assume that the Jacobian matrix $$ DF(u,v)=\begin{pmatrix} \frac{\partial F_1}{\partial u}(u,v) & \frac{\partial F_1}{\partial v}(u,v) \\ \frac{\partial F_2}{\partial u}(u,v) & \frac{\partial F_2}{\partial v}(u,v) \end{pmatrix}$$ has two real and distinct eigenvalues $\lambda(u,v)<\mu(u,v)$. I have seen in the literature that a right eigenvector of $DF(u,v)$ associtated to $\lambda$ ( let's say $r_{\lambda}(u,v)$ ) may be chosen so $r_{\lambda}$ vary smoothly with $(u,v)$ (at least locally), but I don't realize how this can be proved. Thank you for your help in advance!