Reformulating a high-rank linear system into a block-matrix equation

Question

I have on my hands a linear system of equations of the following form $$ \sum_{j=1}^K\sum_{q=1}^N A_{ijpq} x_{jq} = b_{ip} \quad(i=1\dots K,p=1\dots N) $$ in which the $x_{jq}$ are unknown and the rest are given data. I would like to recast this in the form $\tilde A\tilde x=\tilde b$, where $\tilde A$ is a $KN$-by-$KN$ matrix and $\tilde x$ and $\tilde b$ are each vectors of length $KN$. If I fix $K$ and $N$ to concrete values, I can work out by hand what $\tilde A$, $\tilde x$, and $\tilde b$ should be. I am having trouble seeing the generalization for arbitrary $K$ and $N$. Surely there is one, though. What is it?

Answer 1

This turns out to be quite easy to do if one introduces new "super-indices" that run from $1\dots KN$ according to some ordering convention (i.e., row- versus column-major). For instance, defining $$ (ip) = i + K(p-1) \\ (jq) = j + K(q-1) $$ allows the tensor equation in question to be recast from $$ \sum_{j=1}^K\sum_{q=1}^N A_{ijpq}x_{jq} = b_{ip} \qquad (i=1\dots K, p=1\dots N) $$ to the matrix equation $$ \sum_{(jq)=1}^{KN} \tilde A_{(ip),(jq)} \tilde x_{(jq)} = \tilde b_{(ip)} \qquad ((jq)=1..KN) $$ with the correspondence $\tilde A_{(i,p),(j,q)} = A_{ijpq}$, etc.