We propose a model-independent analysis of the neutrino mass matrix through an expansion in terms of the eigenvectors defining the lepton mixing matrix, which we show can be parametrized as small perturbations of the tribimaximal mixing eigenvectors. This approach proves to be powerful and convenient for some aspects of lepton mixing, in particular when studying the sensitivity of the mass matrix elements to departures from their tribimaximal form. In terms of the eigenvector decomposition, the neutrino mass matrix can be understood as originating from a tribimaximal dominant structure with small departures determined by data. By implementing this approach to cases when the neutrino masses originate from different mechanisms, we show that the experimentally observed structure arises very naturally. We thus claim that the observed deviations from the tribimaximal mixing pattern might be interpreted as a possible hint of a ``hybrid nature of the neutrino mass matrix.
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