# Sylvester equation over quaternion

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?

• What do you mean by general meaning? It means Sylvester's equation, not in $M_n(K)$, but in the quaternion algebra. – Dietrich Burde Jan 6 at 19:27
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 $$c$$.