  It is well known that the neutrino mass matrix contains more parameters than
experimentalists can hope to measure in the foreseeable future even if we
impose CP invariance. Thus, various authors have proposed ansatzes to restrict
the form of the neutrino mass matrix further. Here we propose that $m_{\nu_1} +
m_{\nu_2} + m_{\nu_3} = 0$; this ``zero sum'' condition can occur in certain
class of models, such as models whose neutrino mass matrix can be expressed as
commutator of two matrices. With this condition, the absolute neutrino mass can
be obtained in terms of the mass-squared differences. When combined with the
accumulated experimental data this condition predicts two types of mass
hierarchies, with one of them characterized by $m_{\nu_3} \approx -2m_{\nu_1}
\approx -2 m_{\nu_2} \approx 0.063$ eV, and the other by $m_{\nu_1} \approx
-m_{\nu_2} \approx 0.054$ eV and $m_{\nu_3} \approx 0.0064$ eV. The mass ranges
predicted is just below the cosmological upper bound of 0.23 eV from recent
WMAP data and can be probed in the near future. We also point out some
implications for direct laboratory measurement of neutrino masses, and the
neutrino mass matrix.
