How to solve the Sylvester equation $$ax + xb = c$$ over quaternion? I tried to consider operator $$D = a^2 + a\cdot(b+\overline{b}) + b \cdot \overline{b} $$ and calculate $Dx$. But it didn't help.
P.S: What is the general meaning of this equation?
Thank you in advance!
See this paper. I'd interpret $ax+xb=c$ as an equation asking for a quaternion $x=x_0+ x_1i+x_2j+x_3 k$, given quaternions $a$, $b$, and $c$ in a similar way. Write out this equation component wise, and obtain linear equations for the $x_i$. For the existence of a unique solution there will be some assumptions on the parameters $a$, $b$, and $c$.
