Is there a property unique to Kronecker product of two matrices, so that one could use it to prove that a certain matrix has Kronecker product form? Is there a proof technique to this type of objective?
An example problem is the following:Prove that if $P = \{P_i\}$ is the group of all permutation matrices that commute with a particular matrix $A$: $A P_i = P_i A$, and similarly $Q = \{Q_j\}$ has all permutations commuting with $B$, $Q_j B = B Q_j$, then ALL permutations that commute with $A \otimes B$ have Kronecker product form: $$W (A \otimes B) = (A \otimes B) W \quad \Rightarrow \quad W = P_i \otimes Q_j$$
In this example, our goal is to show that $W$ has a Kronecker form. Is there any technique to use for this type of proof?
