I have two complex equations(better to say one).
$$a \times d^* = e^{i k}$$ $$d \times a^* = e^{-i k}$$
where $a^*$ is complex conjugate of $a$.
My attempt:
$x_{1}x_{2}+y_{1}y_{2} = \cos(k)$
$y_{1}x_{2}-x_{1}y_{2} = \sin(k)$
where $a = x_{1} + i y_{1}$ and $d = x_{2} + i y_{2}$
Can this be solved, may not be uniquely?
I am sorry for the naive question, but has already spent lot of time with no avail.
We have: $$ a \cdot \bar d=e^{ik} \; \iff \; a \cdot \bar d \cdot d=e^{ik}\cdot d \; \iff \; a|d|^2=e^{ik}\cdot d \; \iff \;a=\frac{e^{ik}\cdot d }{|d|^2} $$
The two equations are the same and you can solve for $d$ the second equation in the same way.