Solving a matrix equation involving tensors While doing my research, I came across the following matrix equation in $W \in \mathbb{R}^{n \times d}$ that I could not solve.
$$ \sum_{i=0}^{t} X_{i} W Y_{i} + X'_{i}WY'_{i}  = Z $$
where

*

*$X_{i}, X'_{i} \in \mathbb{R}^{n \times d \times n}$


*$Y_{i}, Y'_{i} \in \mathbb{R}^{d \times 1}$


*$Z \in  \mathbb{R}^{n \times d \times 1}$
There is no relation between $t$, $n$ and $d$.
I tried using $WY_{i}$ and $WY'_{i}$ as $K_{i}$ and $J_{i}$ and tried to solve it using least-squares such that I could solve for W from the solution obtained, but I was not getting the right solution ($K_{i}$ and $J_{i}$). Is there any analytical way to solve for W?
 A: $
\def\o{{\tt1}}
\def\bbR#1{{\mathbb R}^{#1}}
\def\qiq{\quad\implies\quad}
\def\LR#1{\left(#1\right)}
\def\shape#1{\operatorname{Reshape}\LR{#1}}
\def\vc#1{\operatorname{vec}\LR{#1}}
$Define a third-order tensor $\nu$ with components
$$\eqalign{
\nu_{\ell jk} &= \begin{cases}
\o\quad{\rm if}\;\;\ell=j+kn-n \\
0\quad{\rm otherwise}
\end{cases}
\qquad
\qquad
\\
}$$
$$\eqalign{
1&\le\; j \;&\le n
 &\qquad\big({\rm the}\;row\;{\rm index}\big) \\
1&\le\; k \;&\le d
 &\qquad\big({\rm the}\;column\;{\rm index}\big) \\
1&\le\; \ell \;&\le n\!\cdot\!d
 &\qquad\big({\rm the}\;long\;{\rm index}\big) \\
}$$
A double-dot product $(:)$ with this tensor will reshape
any matrix $A\in\bbR{n\times d}\,$
into a vector $a\in\bbR{nd\times\o}$
while preserving each component
$$\eqalign{
&a = \nu:A = \vc{A}
 &\qiq a_\ell = \sum_{j=\o}^n\sum_{k=\o}^d \nu_{\ell jk}\,A_{jk} \\
&M_i = \nu:X_i 
 &\qiq (M_i)_{\ell p} = \sum_{j=\o}^n\sum_{k=\o}^d \nu_{\ell jk}\,(X_i)_{jkp} \\
}$$
In the second line, $\nu$ reshaped a tensor $\in\bbR{n\times d\times n}\,$ into a lower rank tensor $\in\bbR{nd\times n}$
Use this tensor to reshape the variables $(X_k,\,X_k^\prime,\,Z)$
into normal matrix and vector variables
$$\eqalign{
M_k &= \nu:X_k,\qquad
N_k &= \nu:X_k^\prime,\qquad
z &= \nu:Z
}$$
Reshape the entire equation into a normal matrix-vector equation,
which can be vectorized to isolate the $W$ matrix as a vector,
which can then be reshaped back into a matrix
$$\eqalign{
&\vc{\sum_k M_kWY_k + N_kWY_k = z} \\
&\LR{\sum_k Y_k^T\otimes M_k + Y_k^T\otimes N_k}w = z \\
&w = \LR{\sum_k Y_k^T\otimes M_k + Y_k^T\otimes N_k}^{-1}z \\
&W = \shape{w,n,d} \;=\; w\cdot \nu \\
}$$
In the last line, note that the $\nu$ tensor can also be used
to reshape the $\,\bbR{nd\times\o}$ vector into
a $\,\bbR{n\times d}$ matrix using a standard dot product.
