# How to rewrite $\left[\begin{smallmatrix} A\otimes B_1\\ \vdots\\ A\otimes B_T \end{smallmatrix}\right]$?

We have the following matrix

$$\left[\begin{array}{c} A\otimes B_1\\ \vdots\\ A\otimes B_T \end{array}\right]$$, where $$A$$ and the $$B_i$$ are matrices, and $$\otimes$$ is the Kronecker product.

Is it possible to rewrite it as a Kronecker product where we gather all the $$B_i$$ matrices as one big matrix?

For example something like $$(\cdots) \otimes \left[\begin{array}{c} B_1\\ \vdots\\ B_T \end{array}\right]$$

You could use the fact that $$\mathbf A \otimes \mathbf B = \mathbf K_{M,N} \cdot \left( \mathbf B \otimes \mathbf A\right) \cdot \mathbf K_{P,Q}^{\rm T},$$ for an $$\mathbf A \in \mathbb{F}^{M \times P}$$, $$\mathbf B \in \mathbb{F}^{N \times Q}$$ and $$\mathbb{F}$$ is the field you are considering.
Here, $$\mathbf K_{M,N}$$ represents the commutation matrix of size $$MN \times MN$$, which is the permutation matrix defined via $$\mathbf K_{M,N}\cdot {\rm vec}\{\mathbf X\} = {\rm vec}\{\mathbf X^{\rm T}\}$$ for any $$\mathbf X \in \mathbb{F}^{M \times N}$$. A lot is known on these and they have some fun properties. Magnus and Neudecker [1,2] would be a good source to study.
You could apply this to each row to "turn around" all your Kronecker products, which then allows to pull out $$\mathbf A$$ completely. This should give you something like a $${\rm diag}\{\mathbf K_1, \ldots, \mathbf K_T\}$$ in front, which may be a bit unhandy. But at least you know that such permutation matrices exist.