Isomorphism $SU(2)/{\pm 1} \leftrightarrow SO(3)$

Question

I have a question regarding the isomorphism from $SU(2)/{\pm 1 \rightarrow SO(3)}$. We defined it to be $\widehat{\Phi(A)x}=A\hat{x}A^{*}$, where $\hat{x}=\sum_{j=1}^3 x_j \sigma_j, x \in \mathbb{R}$ the $\sigma_j$ are the Pauli matrices. My problem is that I don’t see why this is in $SO(3)$. Aren’t $A$ and $x$ 2-dimensional?

Answer 1

Start with a vector $x=(x_1,x_2,x_3)$ in $\mathbf{R}^3$. From that you build the $2\times 2$ complex matrix $\widehat{x}$, and then you compute another $2\times 2$ complex matrix $\widehat{y}=A\widehat{x} A^*$, which turns out to be of the form $\sum_{j=1}^3 y_j \sigma_j$ for some vector $y=(y_1,y_2,y_3)\in \mathbf{R}^3$. That map from $x$ to $y$ (which you have denoted $\Phi(A)$) is in $SO(3)$.