But first, to properly compute anything, it is important to understand what is actually happening during the whole procedure. The SW transformation W = e S being unitary, it does not change the amount of information or the complexity of the Hamiltonian. The resulting shuffle of the matrix elements creates, however, a hierarchy in the information (e.g. eigenvalues), that can be used afterward for a projection in the relevant sector. In addition, when the off-diagonal elements coupling the blocks are much smaller than the typical unperturbed energy scales, a perturbative expansion is allowed to simplify the problem.