I have been thinking of hierarchical block matrix products and how they can relate to a special class of sums of Kronecker products.
Let us say I have a dictionary of matrices $\{M_{1},\cdots,M_N\}$
And a ordered pair of matrices $\{Z,C\}$
$Z$ is the index matrix, with entries in integers $\{0,1,\cdots,N\}$. $C$ is a coefficient matrix over same field as the $M_k$ matrices are.
Consider the matrix
$$\begin{bmatrix}C_{11}\cdot M_{Z_{11}}&C_{12}\cdot M_{Z_{12}} &\cdots\\C_{21}\cdot M_{Z_{21}}& C_{22}\cdot M_{Z_{22}} & \cdots\\\vdots & \ddots &\ddots\end{bmatrix}$$
It should be possible to write as a sum of Kronecker products $$\sum_k ((Z=k)\cdot C)\otimes M_k$$
If we with the (Z=k) notation mean the binary matrix $= \cases{1 , Z_{ij}=k\\0 , Z_{ij}\neq k}$
$\cdot$ being Hadamard product
$\otimes$ being Kronecker product
Now can I prove or disprove that any Kronecker product can be written as such a repeated product, by allowing $M_k$s to be defined in a similar way?
My own work is restricted to imposing the restriction on the dictionary that it must consist of matrices of same dimensionality at any given hierarchical level.
On the lowest possible hierarchical level all matrices can be represented with a trivial dictionary 1x1 matrix with value 1, The C containing the actual matrix and the Z matrix being full of ones.