Solution
= Solution
From $A\sigma_bA^\dagger=R^a{}_b\sigma_a$ and
$\operatorname{tr}(\sigma_a\sigma_b)=2\delta_{ab}$,
$$
\boxed{R^a{}_b=\frac12\operatorname{tr}
(\sigma_aA\sigma_bA^\dagger)}.
$$
= Solution
From $A\sigma_bA^\dagger=R^a{}_b\sigma_a$ and
$\operatorname{tr}(\sigma_a\sigma_b)=2\delta_{ab}$,
$$
\boxed{R^a{}_b=\frac12\operatorname{tr}
(\sigma_aA\sigma_bA^\dagger)}.
$$