I will analyse 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 nu up and nd down singlet vector-like quarks and B) Standard Model with the addition of nR right-handed neutrinos. A discussion of the models developed under this framework in the following papers: 1711.06229 [hep-ph], 1912.05875 [hep-ph] and 2103.13409 [hep-ph], will follow.