# Matrix equation in SL(2,C)

## Main Question or Discussion Point

Hello,

Is the law for matrix multiplication in SL(2,C) the same as usual ? I try to solve the equation $$A_{k}k_{0}A_{k}^{\dagger}=k$$ where $$k_0$$ corresponds to the unit vector $$\{0,0,1\}$$ and $$k$$ is an arbitrary vector, i.e.:

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

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

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 $$a$$):
$$A_{k}k_{0}A_{k}^{\dagger}= \left( \begin{array}{cc} 2aa* & 2ac* \\ 2ca* & 2cc* \\ \end{array} \right)$$

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!

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Fredrik
Staff Emeritus
Gold Member
I'm too lazy to think about the rest, but I multiplied the matrices together and got the same result you did, so that part seems to be OK.

Well, thanks Fredrik but now I really would like to know how to do "the rest" !

gabbagabbahey
Homework Helper
Gold Member
I think you need to know the effect of $A_k$ on at least two linearly independent vectors to completely determine the matrix.

Thanks, actually I found an answer for $$A_{k}$$ without the details of the calculation. I tried to compute $$A_{k}k_{0}A_{k}^{\dagger}$$ but I don't get $$k$$ as would be expected.

The proposed solution is:

$$A_k=\frac{1}{\sqrt{2C(1+n_3)}} \left( \begin{array}{cc} C(1+n_3) & -n_- \\ Cn_+ & 1+n_3 \\ \end{array} \right)$$
i.e.:
$$A_{k}=UB$$
where:
$$U=\frac{1}{\sqrt{2(1+n_3)}} \left( \begin{array}{cc} (1+n_3) & -n_- \\ n_+ & 1+n_3 \\ \end{array} \right)$$
and
$$B= \left( \begin{array}{cc} \sqrt{C} & 0 \\ 0 & \frac{1}{\sqrt{C}} \\ \end{array} \right)$$

The $$C$$ that appears is actually a constant in $$k$$ that I ignored for simplification in my first posting:

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

Can somebody else try to see if this result is correct ?

George Jones
Staff Emeritus
Without further restrictions, I don't think that there is a unique solution for $A_k$. What happens when your $A_k$ is multiplied on the right by an element of the little group of $k_0$?