I have three matrices $A=[a_{ij}]_{n \times n}$ and $B=[b_{ij}]_{t \times n}$ and $V=[v_{ij}]_{n \times m}$ and a vector $C=[c_i]_{n \times 1}$. For $i=1,2,...,n$ the following equations hold,
$\sum_{k=1}^n \sum_{j=1}^t a_{ki} b_{ij} v_{kj} = c_i$
Is there a way that I can summarize these equations into a matrix product form
$XYZ=W$
such that $X, Y$ or $Z$ only contain elements from one of the matrices $A, B$ and $V$ and $W$ only contains elements from vector $C$. It is also fine to pad zeros in either of these matrices but I do not want to impose more restrictions on the elements of matrices $A, B, V$ other than the restrictions posed by the above equations. In other words, I need a matrix form that is exactly equivalent to that set of equations. My guess is, I probably need to use Kronecker multiplication but I don't know how to do it after hours of thinking on it.
Rewriting the sum: \begin{align} c_i &= \sum_{k=1}^n \sum_{j=1}^t a_{ki} b_{ij} v_{kj} \\ &= \sum_{k=1}^n \sum_{j=1}^t b_{ij} v^T_{jk} a_{ki} \\ &= (B V^T A)_{ii} \\ \end{align} So the $c_i$ are the $i$-th diagonal elements of the matrix $BV^TA$. This means that all the multiplications for the non-diagonal elements are wasted. Instead one could use: \begin{align} c_i &= e_i^T (B V^T A) e_i \\ &= (e_i^T B) V^T (A e_i) \\ &= b^T_i V^T a_i \end{align} where $$ a_i = A e_i $$ is the $i$-th column vector of $A$ and $$ b^T_i = e_i^T B \iff B^T e_i = b_o $$ is the $i$-th column vector of $B^T$ which is the $i$-th row vector of $B$.
This is having having $n$ matrix products of the desired form at once.