  We study the minimal seesaw model, where two right-handed Majorana neutrinos
are introduced, focusing on the CP violating phase. In addition, we take the
trimaximal mixing pattern for the neutrino flavor where the charged lepton mass
matrix is diagonal. Thanks to this symmetric framework, the $3\times 2$ Dirac
neutrino mass matrix is given in terms of a few parameters. Numerical studies
reveal that the observation of the CP violating phase can determine the flavor
structure of the Dirac neutrino mass matrix in the minimal seesaw model. In
particular, new minimal Dirac neutrino mass matrices are proposed in the case
of $\rm TM_1$, which is derived by the additional 2-3 family mixing to the
tri-bimaximal mixing basis in the normal hierarchy of neutrino masses. Our
analyses include the Littlest seesaw model by King {\it et al.}, which is one
of the specific one in our results. Furthermore, it is remarked that our
$3\times 2$ Dirac neutrino mass matrix is reproduced by introducing gauge
singlet flavons with the specific alignments of the VEV's. These alignments
suggest the residual symmetry of $S_4$ group.
