# Matrix equation in SL(2,C)

## Homework Statement

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

$$k0= \left( \begin{array}{cc} 2 & 0 \\ 0 & 0 \\ \end{array} \right)$$

$$k= \left( \begin{array}{cc} 1+n_3 & n_- \\ n_+ & 1-n_3 \\ \end{array} \right)$$

## Homework Equations

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

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

## The Attempt at a Solution

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

Thanks a lot for your help!