  We propose a new pattern of the neutrino mixing matrix which can be
parametrized as the product of an arbitrary Hermitian matrix and the well-known
tri-bimaximal mixing matrix. In this scenario, nontrivial values of the
smallest neutrino mixing angle \theta_13 and the CP-violating phases entirely
arise from the non-unitary corrections. We present a complete set of series
expansion formulas for neutrino oscillation probabilities both in vacuum and in
matter of constant density. We do a numerical analysis to show the non-unitary
effects on neutrino oscillations. The possibility of determining small
non-unitary perturbations and CP-violating phases is discussed by measuring
neutrino oscillation probabilities and constructing "deformed unitarity
triangles". Some brief comments on the non-unitary neutrino mixing matrix in
the type-II seesaw models are also given.
