Let
$$X=\begin{bmatrix} A_1 & 0 & ... & 0 \\ 0 & A_2 & ... & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & ... & A_n \end{bmatrix}$$
Where the $A_i$ are square but not necessarily equal in dimension.
Is there any way that $X$ can be expressed using Kronecker products of the $A_i$ matrices?