  Using a sample of 100 H I - selected damped Lyman alpha (DLA) systems,
observed with the High Resolution Echelle Spectrometer on the Keck I telescope,
we present evidence that the scatter in the well-studied correlation between
the redshift and metallicity of a DLA is largely due to the existence of a
mass-metallicity relationship at each redshift. To describe the fundamental
relations that exist between redshift, metallicity and mass, we use a
fundamental plane description, which is described by the following equation:
[M/H]=(-1.9+-0.5)+(0.74+-0.21)logdv_90-(0.32+-0.06)z. Here, we assert that the
velocity width, dv_90, which is defined as the velocity interval containing 90%
of the integrated optical depth, traces the mass of the underlying dark matter
halo. This description provides two significant improvements over the
individual descriptions of the mass-metallicity correlation and
metallicity-redshift correlation. Firstly, the fundamental equation reduces the
scatter around both relationships by about 20%, providing a more stringent
constraint on numerical simulations modeling DLAs. Secondly, it confirms that
the dark matter halos that host DLAs satisfy a mass-metallicity relationship at
each redshift between redshifts 2 through 5.
