I am looking for a more elegant way to confirm the following intuition:
Assume that $A$ and $B$ are two square $p\times p$ matrices. It seems there should always be some matrix $C$ such that $\operatorname{vec}(AB) = C\left[\operatorname{vec}(A)\otimes\operatorname{vec}(B)\right]$.
Is there a more direct way to characterize $C$ (i.e. does it have some well-known name) or some general rule for creating it for a given size or create it for a given $p$?