  We describe the mathematical underpinnings of the biorthogonal von Neumann
method for quantum mechanical simulations (PvB). In particular, we present a
detailed discussion of the important issue of non-orthogonal projection onto
subspaces of biorthogonal bases, and how this differs from orthogonal
projection. We present various representations of the Schr\"odinger equation in
the reduced basis and discuss their relative merits. We conclude with
illustrative examples and a discussion of the outlook and challenges ahead for
the PvB representation.
