Solving linear tensor equation I am sorry if this question has been asked already but I was probably missing the correct keywords.
I am trying to solve a linear equation involving tensors. In Einstein notation it looks like :
$$
F_{di} = K_{dlij} C_{lj}
$$
where $F$ and $K$ are known and we want to know $C$. Also $K$ has the symmetry $K_{dlij} = K_{dlji}$.
For the context, I want to compute the weights of vector-valued finite RKHS.
Any help would be extremely welcome!

I suppose the dimension is $3$, but you can modify correspondingly.
Linearizing indexes in some way, for example
$$
11\to1\\
12\to2\\
13\to3\\
21\to4\\
\ldots
$$
then the equation can be written as
$$
\mathcal{F}_a=\mathcal{K}_{ab}\mathcal{C}_b
$$
where $\mathcal{F},\mathcal{C}$ are vectors in $\mathbb{R}^9$
and if $\mathcal{K}\in\mathbb{R}^9\times\mathbb{R}^9$ is non-singular you can obtain
$$
\mathcal{C}_b=\mathcal{K}^{-1}_{ba}\mathcal{F}_a
$$
then go back to $C$ by
$$
C_{11}=\mathcal{C}_1\\
C_{12}=\mathcal{C}_2\\
C_{13}=\mathcal{C}_3\\
C_{21}=\mathcal{C}_4\\
\ldots
$$
