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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

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    What do you know about $A$,$B$,$C$...$L$? because for instance, if $A$ and $B$ are zero matrices, your thing is easy to solve. It'll all depend on the properties of your matrices2012-11-26
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    Is it a numerical problem (i.e. you look for a solution algorithm) or a theoretical problem (i.e. you look for a closed or semi-closed form solution)? Do you have some specific $A,B,C,I,J,K,L$ in mind?2012-11-26
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    @PatrickDaSilva A,B,C,I,J,K and L are 2x2 non-singular matrices follows properties of ABCD Chain Matrix.2012-11-26
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    @user1551 If i am able to solve this theoretical problem, then i could be able to solve my practical problem.2012-11-26

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