Let $f:\mathbb R^n \to \mathbb R$ be a function that satisfies the equation:
$$\nabla^T f = - (\nabla^T \phi ) f $$ where $\phi:\mathbb R^n \to \mathbb R$ is some given function and $\nabla^T$ is the column partial derivatives operator.
Then obviousely $f(x)=k e^{-\phi(x)}$ solves the equation.
Assume now that we have the equation: $$Q \nabla^T f = - (\nabla^T \phi ) f $$ for some symmetric positive definite matrix $Q$. Can it be solved similarly?