Existing oscillation data point to non-zero neutrino masses with large mixings. We analyze the generic features of the neutrino Majorana mass matrix with inverted hierarchy and construct realistic minimal schemes for the neutrino mass matrix that can explain the large (but not maximal) ν e –ν μ mixing of MSW-LAM as well as the nearly maximal ν μ –ν τ mixing and the small (or negligible) ν e →ν τ transition. These minimal schemes are quite unique and turn out to be extremely predictive. Implications for neutrinoless double beta decay, tritium beta decay and cosmology are analyzed.