Computing the matrix of rotation

Question

Let $\gamma_P$ denote the conjugation by $P\in SU_2$. Let $P=(\cos\theta)I+(\sin\theta)A$ where $A$ is on the equator. I want to know how I compute $\gamma_P$. I know it's defined by conjugation $PBP^*$, but what is $B$?

Answer 1

Given the context, I assume that we're supposed to consider $\gamma_P:\mathfrak{su}_2 \to \mathfrak{su}_2$, where $\mathfrak{su}_2$ denotes the Lie algebra corresponding to $SU_2$, consisting of the traceless skew-Hermitian matrices.

Let $u_1,u_2,u_3$ denote the basis for $SU_2$ given by the Pauli matrices, as described here. That is, we take $$ u_1 = \begin{pmatrix} 0 & i\\ i & 0 \end{pmatrix} \qquad u_2 = \begin{pmatrix} 0 & -1\\ 1 & 0 \end{pmatrix} \qquad u_3 = \begin{pmatrix} i & 0\\ 0 & -i \end{pmatrix} $$ now, compute the matrix of the linear transformation $\gamma_P$ relative to this basis.