  We first obtain the most general and compact parametrization of the unitary
transformation diagonalizing any 3 by 3 hermitian matrix H, as a function of
its elements and eigenvalues. We then study a special class of fermion mass
matrices, defined by the requirement that all of the diagonalizing unitary
matrices (in the up, down, charged lepton and neutrino sectors) contain at
least one mixing angle much smaller than the other two. Our new parametrization
allows us to quickly extract information on the patterns and predictions
emerging from this scheme. In particular we find that the phase difference
between two elements of the two mass matrices (of the sector in question)
controls the generic size of one of the observable fermion mixing angles: i.e.
just fixing that particular phase difference will "predict" the generic value
of one of the mixing angles, irrespective of the value of anything else.
