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How to solve $A_1 \cdot X \cdot A_2+B_1 \cdot X \cdot B_2=C$ for $X$?

I mean, is there a method to convert it into standard form $M \cdot X=N$?

Or other way said, is it possible to express $X$ in terms of $A_1$, $A_2$, $B_1$, $B_2$ and $C$?

I can solve the equation by converting it into system of ordinary linear equations, but want to find a more elegant method.

Widawensen
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azerbajdzan
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  • All matrices are invertible ? If so then just multiply both sides by $A_i^{-1}$ or $B_j^{-1}$ and continue the process to the desired form.. – Widawensen Mar 03 '17 at 19:14
  • @Widawensen: By multiplying by $A_1^{-1}$ I got $X \cdot A_2+A_1^{-1} \cdot B_1 \cdot X \cdot B_2=A_1^{-1} C$. How to proceed? – azerbajdzan Mar 03 '17 at 20:29
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    Try to obtain form $AX+XB=C$ and next use methods from https://en.wikipedia.org/wiki/Sylvester_equation or http://math.stackexchange.com/questions/186598/how-to-solve-matrix-equation-axxb-c-for-x . However I'm not a very specialist of this kind of equations.. so these are only hints.. – Widawensen Mar 04 '17 at 06:08

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