Can this be written more succinctly (preferably as an eigenequation)? I have the set of equations $$\pmatrix{L_x \\\ L_y \\\ L_z}=\pmatrix{\partial_xL_x &\partial_yL_x &\partial_zL_x \\\ \partial_xL_y &\partial_yL_y &\partial_zL_y \\\ \partial_xL_z &\partial_yL_z &\partial_zL_z} \pmatrix{x \\\ y \\\ z}$$ I feel like this should be expressible in terms of the vectors $$L=\pmatrix{L_x \\\ L_y \\\ L_z}\\!, \;\;r=\pmatrix{x \\\ y \\\ z}\\!,$$ some differential operators (like the gradient or the divergence) and perhaps a matrix with some zeros and ones, but I've been unable to come up with anything yet. **Any ideas?** Ideally one would be able to write it like some vector DE whose solutions could be said something intelligent about, but this may not be possible.

The equations for $L_x$, $L_y$ and $L_z$ are independent and equal. Writing $L_x = u$, you obtain \begin{equation} u = r \cdot\
abla u = x\, u_x + y\, u_y + z\, u_z. \end{equation} You can easily solve this first order linear partial differential equation using the method of characteristics.