I am having a algebric problem in my thesis work. It is some how like this ...
I have to find $X$, $Y$, $X'$ and $Y'$, where these are unknown $2\times 2$-matrices and $A$, $B$, $C$, $I$, $J$, $K$ and $L$ are known $2\times 2$-matrices. \begin{align*} A \cdot X \cdot Y \cdot B &= I\\ A \cdot X \cdot Y' \cdot B &= J\\ A \cdot X \cdot Y \cdot C \cdot X' \cdot Y' \cdot B &= K\\ A \cdot X' \cdot Y' \cdot B &= L \end{align*} Real goal was to find $X$ and $Y$ matrices (individually), more equations are created to simplify problem and make knowns and unknowns equal.
It is somehow looks realistic, because right now I have 4 equations and 4 unknowns. Further equations can be generated by keeping 2 unknowns between $A$ and $B$.
Please can anyone say about it? Thanks