I have a block matrix
$$X=\left[ \begin{matrix} {{X}_{1,1}} & {{X}_{1,2}} & \cdots & {{X}_{1,M}} \\ {{X}_{2,1}} & {{X}_{2,2}} & \cdots & {{X}_{2,M}} \\ \vdots & \vdots & {} & \vdots \\ {{X}_{N,1}} & {{X}_{N,2}} & {} & {{X}_{N,M}} \\ \end{matrix} \right]$$ where each ${{X}_{i,j}}$ is a block. The size of $X$ is $n \times m$, with $N \times M$ blocks. I also have a $N \times M$ matrix: $$Y=\left[ \begin{matrix} {{y}_{1,1}} & {{y}_{1,2}} & \cdots & {{y}_{1,M}} \\ {{y}_{2,1}} & {{y}_{2,2}} & \cdots & {{y}_{2,M}} \\ \vdots & \vdots & {} & \vdots \\ {{y}_{N,1}} & {{y}_{N,2}} & \cdots & {{y}_{N,M}} \\ \end{matrix} \right]$$ where each ${{y}_{i,j}}$ is a scalar.
Now, how can I use matrix $Y$ and $X$, together with some operator (e.g. element-wise multiplication), to represent the following matrix:
$$\left[ \begin{matrix} {{y}_{1,1}}{{X}_{1,1}} & {{y}_{1,2}}{{X}_{1,2}} & \cdots & {{y}_{1,M}}{{X}_{1,M}} \\ {{y}_{2,1}}{{X}_{2,1}} & {{y}_{2,2}}{{X}_{2,2}} & \cdots & {{y}_{2,M}}{{X}_{2,M}} \\ \vdots & \vdots & {} & \vdots \\ {{y}_{N,1}}{{X}_{N,1}} & {{y}_{N,2}}{{X}_{N,2}} & \cdots & {{y}_{N,M}}{{X}_{N,M}} \\ \end{matrix} \right]$$
First, let $A$ be an $n\times m$ matrix of ones. Then $$ X \cdot (Y \otimes A) $$ gives your desired matrix where $\otimes$ denotes the Kronecker product, and $\cdot$ denotes the Hadamard product.