Let $A=[a_{ij}]$ be an $m \times m$ matrix and $B$ be a $nm \times nm$ block diagonal matrix with diagonal blocks $B_i$, $i=1, \ldots, m$, $n \times n$ matrices. I want to express

$\left[ \begin{array}[cccc] \\a_{11} B_1 & a_{12}B_1 & \cdots & a_{1m} B_1 \\ a_{21} B_2 & a_{22}B_2 & \cdots & a_{2m} B_2 \\ \vdots & \vdots & \vdots & \vdots\\ a_{m1} B_m & a_{m2}B_m & \cdots & a_{mm} B_m \\ \end{array}\right]_{mn \times mn}$

in a compact form, possibly in one shot using Kronecker products. Is it possible?