What's the Jacobian of the sign function for vectors or:
$$A = \frac{\mathrm{d} \hat{v}}{\mathrm{d} \vec{v}}$$
I think it is probably some kind of dirac delta or something like:
$$A\vec{u} = 2\delta{\left\vert\vec{v}\right\rvert}\vec{u} $$
but I'm not sure. I think there should be some kind of cross product there but using the vector $\left(1, 1, 1\right)$ seems wrong to me.
Let ${\hat v}$ denote a unit vector, and let ${\vec v}$ denote a vector in the direction of ${\hat v}$ but with variable length $\lambda$.
Find the differential of ${\vec v}$ $$\eqalign{ {\vec v} &= \lambda\,{\hat v} \cr d{\vec v} &= {\hat v}\,d\lambda + \lambda\,d{\hat v} \cr d{\hat v} &= \frac{d{\vec v} - {\hat v}\,d\lambda}{\lambda} \cr &= \frac{\lambda^2d{\vec v} - {\vec v}\lambda\,d\lambda}{\lambda^3} \cr }$$ We also have the relation $$\eqalign{ \lambda^2 &= {\vec v}^T{\vec v} \cr \lambda\,d\lambda &= {\vec v}^Td{\vec v} \cr\cr }$$ Substituting this into the previous result yields
$$\eqalign{ d{\hat v} &= \frac{\lambda^2Id{\vec v} - {\vec v}{\vec v}^Td{\vec v}}{\lambda^3} \cr \frac{d{\hat v}}{d{\vec v}} &= \frac{\lambda^2I - {\vec v}{\vec v}^T}{\lambda^3} \cr &= \frac{I - {\hat v}{\hat v}^T}{\lambda} \cr}$$