Sylvester equation over quaternion












2












$begingroup$


How to solve the Sylvester equation $$ax + xb = c$$ over quaternion? I tried to consider operator $$D = a^2 + acdot(b+overline{b}) + b cdot overline{b} $$ and calculate $Dx$. But it didn't help.



P.S: What is the general meaning of this equation?



Thank you in advance!






abstract-algebra quaternions sylvester-equation







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
















2












$begingroup$


How to solve the Sylvester equation $$ax + xb = c$$ over quaternion? I tried to consider operator $$D = a^2 + acdot(b+overline{b}) + b cdot overline{b} $$ and calculate $Dx$. But it didn't help.



P.S: What is the general meaning of this equation?



Thank you in advance!






abstract-algebra quaternions sylvester-equation







share|cite|improve this question









$endgroup$












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














2












2








2





$begingroup$


How to solve the Sylvester equation $$ax + xb = c$$ over quaternion? I tried to consider operator $$D = a^2 + acdot(b+overline{b}) + b cdot overline{b} $$ and calculate $Dx$. But it didn't help.



P.S: What is the general meaning of this equation?



Thank you in advance!






abstract-algebra quaternions sylvester-equation







share|cite|improve this question









$endgroup$




How to solve the Sylvester equation $$ax + xb = c$$ over quaternion? I tried to consider operator $$D = a^2 + acdot(b+overline{b}) + b cdot overline{b} $$ and calculate $Dx$. But it didn't help.



P.S: What is the general meaning of this equation?



Thank you in advance!






abstract-algebra quaternions sylvester-equation




abstract-algebra quaternions sylvester-equation






share|cite|improve this question













share|cite|improve this question











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asked Jan 6 at 14:01









mathmaniacmathmaniac

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18011












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


















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
















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




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










1 Answer
1






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oldest

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2












$begingroup$

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 $c$.






share|cite|improve this answer









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    $begingroup$

    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 $c$.






    share|cite|improve this answer









    $endgroup$


















      2












      $begingroup$

      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 $c$.






      share|cite|improve this answer









      $endgroup$
















        2












        2








        2





        $begingroup$

        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 $c$.






        share|cite|improve this answer









        $endgroup$



        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 $c$.







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered Jan 6 at 19:19









        Christian BlatterChristian Blatter

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