  We explore the robustness of the correlation matrix Hamiltonian
reconstruction technique with respect to the choice of operator basis, studying
the effects of bases that are undercomplete and over-complete -- too few or too
many operators respectively. An approximation scheme for reconstructing from an
undercomplete basis is proposed and performed numerically on select models. We
discuss the confounding effects of conserved quantities and symmetries on
reconstruction attempts. We apply these considerations to a variety of
one-dimensional systems in zero- and finite-temperature regimes.
