Questions about polynomial algbras

Question

If $x$ is vector of dimension 3, say $(x_1,x_2,x_3)^T$, then $S^d$ is an operator such that $$ S^d(x)=\left( \begin{matrix} x_2 & x_1 & 0 &0 & 0& ...& 0 &0&0\\ x_3 & x_2 & x_1 & 0 & 0& ...& 0 &0&0\\ 0 & x_3 & x_2 & x_1& 0 &...& 0 &0&0\\ 0 & 0 & x_3 & x_2& x_1 &...& 0 &0&0\\ 0 & 0 & 0 & x_3& x_2 &...& 0 &0&0\\ 0 & 0 & 0 & 0& x_3 &...& 0 &0&0\\ ...&...&...&...&...&...&...&...&...\\ 0 & 0 & 0 & 0& 0 &...& x_3 &x_2&x_1\\ 0 & 0 & 0 & 0& 0 &...& 0 &x_3&x_2\\ \end{matrix} \right) $$ The matrix $S^d(x)$ is $d \times d$ matrix $(d>3)$ where $S^d_{ij}=x_{i-j+2}$ if $0<(i-j+2)<4$ and $S^d_{ij}=0$ otherwise. My question is for any given $d \times d$ matrix A, can I find a finite sequence $\{x^i\}_{i=1}^N$ where $x^i$ is a vector of dimension 3, such that $$A=S^d(x^1)\times S^d(x^2)\times ...\times S^d(x^N) $$\ And if could, please provide some bound of N. \ Actually, this is related to the convolution neural network and I am trying to prove that the fully connected network belongs to the convolution neural network.