Geometry of the canonical Van Vleck transformation

A Van Vleck transformation U=e^g; g=-g^+, to an effective Hamiltonian changes an orthonormal basis in the zeroth order eigenspace Omega_0 to one in the subspace Omega of the corresponding exact eigenvectors. The canonical U_c=e^g_c is the only where g is odd. Joergensen's, theorem of uniqueness reveals that U_c equals U_c'=P(P_0PP_0)^(-1/2)+Q(Q_0QQ_0)^(-1/2) where P_0/P project on Omega_0/Omega, and where Q_0=1-P_0 and Q=1-P. By Klein's theorem of uniqueness, u_c=P(P_0PP_0) is the mapping which changes an orthonormal basis in Omega_0 minimally. In the present paper, Klein's theorem is developed, proven by simple geometry and also as a direct consequence of Joergensen's. It is shown that U_c' equals |S|^(-1)S=S|S|^(-1) where S=PP_0+QQ_0 satisfies SS^+=S^+S=|S|^2. These commutations simplify earlier proofs, lead to g_c in terms of P_0 and P and to a series of geometrical interpretations, all easily illustrated in the elementary case where Omega_0 and Omega are 2-dimensional planes in the 3-dimensional space. Thus U_c: Omega_0-->Omega is the reflection in the plane Omega_m between Omega_0 and Omega as well as a rotation around their line of intersection.