Matrix equation in SL(2,C)

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Homework Statement



Solve the equation [tex]A_{k}k_{0}A_{k}^{\dagger}=k[/tex] in SL(2,C) where [tex]k_0[/tex] corresponds to the unit vector [tex]\{0,0,1\}[/tex] and [tex]k[/tex] is an arbitrary vector, i.e.:

[tex]k0=
\left( \begin{array}{cc}
2 & 0 \\
0 & 0 \\
\end{array} \right)
[/tex]

[tex]k=
\left( \begin{array}{cc}
1+n_3 & n_- \\
n_+ & 1-n_3 \\
\end{array} \right)
[/tex]



Homework Equations



If I try to solve for
[tex]A_k=
\left( \begin{array}{cc}
a & b \\
c & d \\
\end{array} \right)
[/tex]

this gives (where [tex]a*[/tex] is the conjugate of [tex]a[/tex]):
[tex]A_{k}k_{0}A_{k}^{\dagger}=
\left( \begin{array}{cc}
2aa* & 2ac* \\
2ca* & 2cc* \\
\end{array} \right)
[/tex]



The Attempt at a Solution



So this gives conditions on [tex]\{a,c\}[/tex] but can [tex]\{b,c\}[/tex] be arbitrary ? How do I solve this equation and obtain the expression of [tex]A_k[/tex] involving only [tex]n_+, n_-[/tex] and [tex]n_3[/tex] ?

Thanks a lot for your help!

Homework Statement





Homework Equations





The Attempt at a Solution

 

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