  This work analyses a class of extensions of the Standard Model through the
lens of a novel exact parameterisation. This parameterisation is specially
suitable to the analysis of extensions of the Standard Model with non-unitary
mixing matrices, like models with vector-like fermions or right-handed
neutrinos. The usefulness of this parameterisation is motivated with two
example models: A) Standard Model with the addition of $n_u$ up and $n_d$ down
singlet vector-like quarks and B) Standard Model with the addition of $n_R$
right-handed neutrinos.
