Trace of a matrix equation

Question

If I have a matrix equation like

$$e = \operatorname{sign}(Bx - Cd) : (Bx - Cd)$$

were $B$ and $C$ are matrix and $x$ and $d$ are vectores, and I need to find the gradient of e in sense of x, $\nabla e$. How can I take this derivative?

Answer 1

Let $M=(Bx-Cd)$
then the function, differential, and gradient are (respectively) $$\eqalign{ e &= {\rm sign}(M):M \cr de &= {\rm sign}(M):dM = {\rm sign}(M):B\,dx = B^T{\rm sign}(M):dx \cr \nabla e = \frac{\partial e}{\partial x} &= B^T{\rm sign}(M) \cr\cr }$$ The gradient is undefined wherever a component of $M$ is equal to zero.