It's a problem from 18.065 ocw assignment, problem description is shown in the block below:
If an invertible matrix $X$ satisfies the Sylvester equation $AX − XB = C$, find a Sylvester equation for $X^{−1}$.
Although I know that matrices $A$ and $B$ are normal, I still couldn't derive a formula for $X^{-1}$. Is there anybody having any tip about this problem?
Upadte: I misunderstood the problem at the first time, it is asking a Sylvester equation for inverse, not the inverse itself.
$$AX - XB = C$$ Multiplying both sides by $X^{-1}$ on the left and right: $$X^{-1}(AX - XB)X^{-1} = X^{-1}CX^{-1}$$ $$X^{-1}A - BX^{-1} = X^{-1}CX^{-1}$$ Rearrange the position: $$- BX^{-1} + X^{-1}A = X^{-1}CX^{-1}$$
Hint: Multiply both sides of the equation $AX-XB = C$ by $X^{-1}$ on the left and $X^{-1}$ on the right. What do you get when you simplify things?