If $C=A\otimes B$, the Kronecker product of two matrices, how do you determine $A$ and $B$

Question

If $A$ is an $m \times n$ matrix and $B$ is a $p \times q$ matrix, then the Kronecker product $C = A \otimes B$ is an $mp \times nq$ matrix. Assume that I have a matrix $C$ and that I want to find the matrices $A$ and $B$ such that $C=A\otimes B$, how can I find them?

Answer 1

Firstly, they aren't uniquely determined, consider $C=[0]_{m,n}$ and any $m$ by $n$ matrix $A$, then we have $$A\otimes [0]_{1,1}=C.$$

If you want to recover some $A,B$ such that $A\otimes B = C$, then you can take $C=C\otimes [1]$ (meaning the identity matrix in $M_1(R)$).