The 2D rotation matrix \(R(\phi)\) defined in Eq. (6.3) is a reducible representation of \(\mathrm{SO}(2)\). Diagonalize \(R(\phi)\) to give the eigenvalues \(\lambda_{ \pm}=e^{ \pm i \phi}\) and Eq. (6.11). Show that the basis vectors in the new basis after diagonalization are given by Eq. (6.12). Find the form of the generator \(J\) given by Eq. (6.10) in the new basis. Find the operators \(C\) and \(C^{-1}\) that perform the similarity transformation between the original basis and the diagonalized basis, \(C R(\phi) C^{-1}=R^{\prime}\). Hint: The solution of Problem 14.14 gives much of the required math.