First off, I'm not very experienced with the subject and English is also not my first language, so if there are any inaccuracies in the following text, let me know.
Given a linear, scalar, second-order PDE $$ a(x,y) u_{xx} + 2b(x,y) u_{xy} + c(x,y) u_{yy} = f(x,y,u_x, u_y, u) $$ and an invertible, $\ C^2 $ coordinate transformation $$\phi(x,y) = (\xi (x,y), \eta (x,y)), $$ I'm supposed to show that this transformation projects characteristic curves onto characteristic curves of the transformed PDE.
What I did so far: I transformed the PDE via chain-rule, and got the result that if $\ (x(s), y(s)) $ is my characteristic curve, i.e.
$$ \begin{pmatrix} x' & y' \end{pmatrix} \begin{pmatrix} a & b \\ b & c \end{pmatrix} \begin{pmatrix} x' \\ y' \end{pmatrix} = 0, $$ then what I need to show for the transformed curve $\ (\xi (x(s), y(s)), \eta (x(s), y(s)) ) $ is
$$ \begin{pmatrix} \xi' & \eta' \end{pmatrix} \begin{pmatrix} a \xi^2_x + 2b \xi_x \xi_y + c \xi_y^2 & a \xi_x \eta_x + b(\xi_x \eta_y + \eta_x \xi_y) + c \xi_y \eta_y \\ a \xi_x \eta_x + b(\xi_x \eta_y + \eta_x \xi_y) + c \xi_y \eta_y & a\eta^2_x + 2b \eta_x \eta_y + c \eta_y^2 \end{pmatrix} \begin{pmatrix} \xi' \\ \eta' \end{pmatrix} = 0, $$ or in other words, and again with the chain-rule
$$\begin{pmatrix} \xi' & \eta' \end{pmatrix} \begin{pmatrix} \xi_x & \xi_y \\ \eta_x & \eta_y \\ \end{pmatrix} \begin{pmatrix} a & b \\ b & c \\ \end{pmatrix} \begin{pmatrix} \xi_x & \eta_x \\ \xi_y & \eta_y \\ \end{pmatrix} \begin{pmatrix} \xi' \\ \eta' \end{pmatrix} = \\ = \begin{pmatrix} x' & y' \end{pmatrix} \begin{pmatrix} \xi_x & \eta_x \\ \xi_y & \eta_y \\ \end{pmatrix} \begin{pmatrix} \xi_x & \xi_y \\ \eta_x & \eta_y \\ \end{pmatrix} \begin{pmatrix} a & b \\ b & c \\ \end{pmatrix} \begin{pmatrix} \xi_x & \eta_x \\ \xi_y & \eta_y \\ \end{pmatrix} \begin{pmatrix} \xi_x & \xi_y \\ \eta_x & \eta_y \\ \end{pmatrix} \begin{pmatrix} x' \\ y' \end{pmatrix} = 0. $$
This is where it ends for me, I don't know how to show that last equality. And I have my doubts that everything I've done up to that point is absolutely correct. I would just like to know if there is a severe understanding problem from my side and if I've made a terrible mistake somewhere. Any help is appreciated.