i want to solve the following equation. $\begin{bmatrix} \mathbf{A_1} & \mathbf{A_2} & \mathbf{A_3} & \mathbf{A_4} \\ \mathbf{A_5} & \mathbf{A_6} & \mathbf{A_7} & \mathbf{A_8} \end{bmatrix} \begin{bmatrix} \mathbf{w_1} \\ \mathbf{w_2} \\ \mathbf{w_3} \\ \mathbf{w_4}\end{bmatrix} = \mathbf{0}$ say Equation (1)
where $\mathbf{A}_i = \mathbf{B}_i \otimes \mathbf{D}_i$. Here, $\mathbf{B}_i$ and $\mathbf{D}_i$ are matrice of size 2 x 2 with non-zero elements and full rank. These matrices are given.
$\mathbf{w}_i = \mathbf{x}_i \otimes \mathbf{y}_i$. Here, $\mathbf{x}_i$ and $\mathbf{y}_i$ are vectors of size 2 x 1 and are variables. How to find the vectors $\mathbf{x}_i$ and $\mathbf{y}_i$?
Note: I calculated the solution space of the $\mathbf{w}$ vectors by calculating the null space of the matrix. i took one vector from the solution space and tried to find $\mathbf{x}_i$ and $\mathbf{y}_i$ from $\mathbf{w}_i$ but it didnot satisfy $\mathbf{w}_i = \mathbf{x}_i \otimes \mathbf{y}_i$.
Thank you in advance