[Here](https://mathoverflow.net/questions/371010/how-to-calculate-inverse-of-sum-of-two-kronecker-products-with-specific-form-eff) and [here](https://www.sciencedirect.com/science/article/pii/S0024379587903144?via%3Dihub), specific ways to address the equation in $x$, for $N=2$, are given:


$$\sum\_{i=1}^N (A\_i\otimes B\_i)x=c$$


Is anything know about the case $N>2$?


I am looking in fact for an efficient solution to the above type of linear system. Such structure may arise from space-time algorithms applied to parabolic, non-linear problems.




---


In fact, the system I am interested in has the following structure:


$\sum\_{j,l,a,b}M\_{j,a}^iT\_{l,b}^k\nu^1\_{a,b}x^1\_{j,l}+\sum\_{j,l,a,b,o}S\_{j,a}^iD\_{l,b,o,j}^k\nu^2\_{a,b} x^2\_{j,l}=f^{i,k}$


As you can see, the $\nu$ factors prevent me from writing the system as in the link I posted. Also, the matrix $D$ contains the index $j$, which gives further problems. My idea was to decompose $\nu$ as a sum of Kronecker products (approximately, and carry out a similar procedure for $D$, I won't go into details), hoping to obtain a better structure.


By doing so, we see that we obtain the original system I decsribed above. So, to answer to @Nathaniel, $N$ should be large so that the approximation of e.g. $\nu$ in terms of a sum of Kronecker product, is good. I don't have a specific number in mind.


As for the size of $A\_i, B\_i$, they are square, of side $1e4$, $1e3$ respectively. I suppose then that $N$ will be much smaller than these numbers.



Linear system with sum of Kronecker products
The recent state of the art is described in section 7.2 of Simoncini, V. "Computational methods for linear matrix equations." *SIAM Rev.* **58**, 377 (2016), <https://doi.org/10.1137/130912839>. Your equation is equivalent to equation (2) in that reference. A lightly reformatted quote from there:



> 
> Equation (2) is very difficult to analyze in its full generality, and
> necessary and sufficient conditions for the existence and uniqueness
> of the solution explicitly based on $\{A\_i\}$, $\{B\_i\}$ are hard to
> find, except for some very special cases.
> 
> 
> 


