Given a set of $m$-dimensional complex vectors $\{v_{11},v_{12},..,v_{nn}\}$, does there exist a set of $m\times m$ complex matrices $\{F_1,F_2,...,F_n\}$ such that
\begin{equation} (F_a)_{ij} = \sum_k (F_b)_{ik}(v_{ij})_k(F_c)_{kj} \end{equation} holds for all $a,b,c\leq n$. This system of equations pops up as a consistency condition in some algebraic/category theoretic framework. These equations are fairly general and I do not expect a full answer to exist, but I am interested in the more general structure of these equations.
The above equations look like some non-standard matrix product form, where if all $(v_{ij})_k = 1$ would result in $F_a = F_b F_c$. Does this form exist in the literature and if so are there any useful properties of this non-standard product? More interestingly can we write this form in a more standard linear algebra form in terms of the matrices?