Population analyses have become an indispensable tool to computational
chemists. Yet implementation within popular quantum chemistry software
has buried the interesting philosophical choices made when partitioning
the electron density into atomic contributions. There is further
historical context that has significantly influenced common conceptions
of chemical bonding and reactivity. This work reviews select aspects of
orbital and spatial decomposition schemes of the density matrix,
pointing out essential linear algebraic considerations and associated
tools of shared interest to us and Prof. Mayer.