Permute the factors of a Kronecker product

Question

Let two matrices $A$ and $B$ of size $m\times n$ and $p \times q$, respectively.

What is the expression of two matrices $F$ and $G$ such that $A \otimes B = F ( B \otimes A ) G$?

Answer 1

The matrices $F$ and $G$ are called commutation matrices. A commutation matrix $K_{m,n}$ is the unique permutation matrix that satisfies $K_{m,n} \cdot {\rm vec}\{X\} = {\rm vec}\{X^T\}$ for any $X$ of size $m \times n$, where ${\rm vec}\{\cdot\}$ is the vectorization operator that stacks $X$ into a vector (column by column). These commutation matrices can be used to permute Kronecker products. They satisfy $$K_{m,p}^T \cdot(A \otimes B) \cdot K_{n,q} = B \otimes A,$$ where $A$ and $B$ are $m \times n$ and $p\times q$. Hence, your $F$ is $K_{m,p}$ and your $G$ is $K_{n,q}^T$. You can find a lot more details and properties about these matrices in [MN79, MN95].

[MN95] Magnus, Jan R.; Neudecker, Heinz, Matrix differential calculus with applications in statistics and econometrics, ZBL07044055, 1995.

[MN79] Magnus, Jan R.; Neudecker, H., The commutation matrix: Some properties and applications, Ann. Stat. 7, 381-394 (1979). ZBL0414.62040.