I am trying to code this problem: Suppose there are four elements to use when tensoring: $$ \left\{ I = \begin{bmatrix}1&0\\0&1\end{bmatrix},\, X = \begin{bmatrix}0&1\\1&0\end{bmatrix},\, Y = \begin{bmatrix}0&-i\\i&0\end{bmatrix},\, Z = \begin{bmatrix}1&0\\0&-1\end{bmatrix} \right\}, $$
where $i = \sqrt{-1}$.
The tensor product is denoted by $\otimes$ and applied on two matrices $A_{n \times n}$ and $B_{m \times m}$ as such: $$ A\otimes B = \begin{align} \begin{bmatrix} A_{11}B & A_{12}B & \ldots & A_{1n}B\\ A_{21}B & A_{22}B & \ldots & A_{2n}B\\ \vdots & \vdots & \ddots & \vdots \\ A_{n1}B & A_{n2}B & \ldots & A_{nn}B\\ \end{bmatrix}_{nm\times nm} \end{align} $$
You are given the following: $$ \begin{align} A_1 &= a_{11} \otimes a_{12} \otimes \cdots \otimes a_{1n}\\ A_2 &= a_{21} \otimes a_{22} \otimes \cdots \otimes a_{2n}\\ &\phantom{~~}\vdots\\ A_m &= a_{n1} \otimes a_{m2} \otimes \cdots \otimes a_{mn}\\ \end{align}, $$ where $a_{ij} \in \{I, X, Y, Z\}$.
Let $A = A_1 * A_2 * \cdots * A_m$, where $*$ is ordinary left-to-right matrix multiplication. I would assume there is some tensor product combination that equals $A$, i.e., $$ A = a_{1} \otimes a_{2} \otimes \cdots \otimes a_{n}, \text{ where } a_i \in \{I,\, X,\, Y,\, Z\}.\\ $$ How can I find the tensor product combination of $A$? Please keep in mind that I don't know if my assumption is correct. Furthermore, there are similar questions that have been asked such as "Inverse" of tensor product and The Inverse of a Fourth Order Tensor but the details and requirements of my question differ.