Math genius here. I have this problem of vectorizing the block matrix of bilinear entries as follows: $$W=\begin{bmatrix} e_1M_{11}e_1^T & e_1M_{12}e_2^T\\e_2M_{21}e_1^T&e_2M_{22}e_2^T \end{bmatrix}$$
Is it possible to write the matrix $W$ as the product of some matrics whose entries are $e_1,e_2, M_{11}$ and so on.? Note that $e_1,e_2$ are vectors and $M_{ij}$ are matrics of compatible shapes.
Yes: you can write it as
$$ \begin{bmatrix} e_{1} & 0 \\ 0 & e_{2} \end{bmatrix} \begin{bmatrix} M_{11} & M_{12}\\ M_{21} & M_{22} \end{bmatrix} \begin{bmatrix} e_{1}^{T} & 0\\ 0 & e_{2}^{T} \end{bmatrix} $$ (where the zeros are blocks or the appropriate size).