Can this be written more succinctly (preferably as an eigenequation)?

Question

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.

Answer 1

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\nabla 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.