Does there exist a necessary and sufficient condition when four complex numbers $c_{11},c_{12},c_{21},c_{22}$ can be expressed as
$$ \begin{pmatrix} c_{11}\\c_{12}\\c_{21}\\c_{22} \end{pmatrix} = \begin{pmatrix} c_{1}d_{1}\\c_{1}d_{2}\\c_{2}d_{1}\\c_{2}d_{2} \end{pmatrix} $$
where $c_{1},c_{2},d_{1},d_{2}$ are four complex numbers?
In other words, is there a necessary and sufficient condition when $$ \begin{pmatrix} c_{11}\\c_{12}\\c_{21}\\c_{22} \end{pmatrix} = \begin{pmatrix} c_{1}\\c_{2}\\ \end{pmatrix} \otimes \begin{pmatrix} d_{1}\\d_{2}\\ \end{pmatrix} $$
where $\otimes$ is the Kroneceker product?
P.S. - I welcome answers from any field; answers don't have to originate from linear algebra and tensor products (but it's good if they do :) ). I just want to know when a four-tuple can be expressed as certain products.
A necessary condition is $$c_{11}c_{22}=c_{12}c_{21}.$$
It is also sufficient. Let's start with the case of some $c_{ij}=0$. If $c_{11}=0$ then $c_{12}=0$ or $c_{21}=0$. In the first case set $c_1=0$, otherwise set $d_1=0$ and now you can surely solve the other $2$ equations.
If the $c_{ij}$ are all different from $0$, set $c_1=1$, $d_1=c_{11}$, $d_2=c_{12}$. You have to solve $$c_{21}=c_2c_{11}$$ $$c_{22}=c_2c_{12}.$$ Lucky for us, $$\frac{c_{21}}{c_{11}}=\frac{c_{22}}{c_{12}}.$$