Superficially simple vector differential equation problem

cfp

Hi,

I have the following vector differential equation (numerator layout derivatives):

[tex]\frac{\partial e(v)}{\partial v}=\frac{1}{\beta} \frac{\partial w(v)}{\partial v} \Gamma^{-1}[/tex]

where both ##e(v)## and ##w(v)## are scalar functions of the vector ##v##, and where ##\Gamma## is a symmetric invertible matrix with all columns (and rows) summing to 1.

The naive solution would be ##e(v)=\frac{1}{\beta} w(v) \Gamma^{-1}##, but this is incorrect since ##e(v)## is a scalar.

Clearly, when ##\Gamma## is the identity matrix, ##e(v)=\frac{1}{\beta} w(v)## is a valid solution. My question is, does a solution exist for any other value of ##\Gamma##?

Tom