I'm trying to find a complex number $c$ that minimizes $\Vert \mathbf{v}-c\mathbf{z}\Vert$ for vectors $\mathbf{v}$ and $\mathbf{z}$ in $\mathbb{C}^n$.
I guess I'm looking for $$\frac{\partial}{\partial c} \Vert \mathbf{v}-c\mathbf{z}\Vert = 0$$
The squared norm is easier to deal with, I think. Either way, when you evaluate this, you end up trying to differentiate $\bar{c}$, which of course is meaningless.
This answer covers the general problem. Unfortunately it puts things in terms of holomorphic coordinates, an idea I'm unfamiliar with. There doesn't seem to be a good description of them online either. Wikipedia gets into atlases, charts, and varieties, leading me deeper into a rabbit hole I think probably isn't necessary.
Help?
Squaring the norm gives minimization of $$ \|v-cz\|^2=\|v\|^2-2\text{Re}\,\bar{v}^Tcz+|c|^2\|z\|^2. $$ Write $c=re^{i\theta}$ and minimize the middle term with respect to $e^{i\theta}$ (taking $\theta=-\arg(\bar{v}^Tz)$ will make the real part to be $r|\bar{v}^Tz|$), then use the real $r$ to minimize the quadratic polynomial.