Kronecker product rule

Question

Let $A\in\mathbb{R}^{p\times q}$, $B\in\mathbb{R}^{r\times s}$, and $C\in\mathbb{R}^{pr\times qs}$. Consider: \begin{equation} (A\otimes B)C^{\top}. \end{equation} I'm wondering if there are any rules for how one can compute the above expression. I initially thought that the following would hold: \begin{equation} (A\otimes B)C^{\top}=(AC^{\top}\otimes B), \end{equation} however, $AC^{\top}$ is not a well-defined matrix product per dimensions. So what can one do instead?

Answer 1

One common trick is to write $C$ as a sum of the form $$ C = \sum_{i=1}^n P_i \otimes Q_i $$ With each $P_i$ of size $p\times q$ and each $Q_i$ of size $r \times s$. With that, we can rewrite $$ (A \otimes B)C^{\top} = (A \otimes B)\left(\sum_{i=1}^n P_i \otimes Q_i\right)^{\top} = \sum_{i=1}^n (AP_{i}^{\top}) \otimes (BQ_{i}^{\top}). $$ If we take $P_i$ to be the standard basis of matrices over $\Bbb R^{p \times q}$ (i.e. the matrices with exactly one non-zero entry), then writing $C$ as this sum corresponds to breaking it down into block-entries.