For decades, researchers have been trying to understand how people form their opinions. This quest has become even more pressing with the widespread usage of online social networks and social media, which seem to amplify the already existing phenomenon of polarization. In this work, we study the problem of polarization assuming that opinions evolve according to the popular Friedkin-Johnsen (FJ) model. The FJ model is one of the few existing opinion dynamics models that has been validated on small/medium-sized social groups. First, we carry out a comprehensive survey of the FJ model in the literature (distinguishing its main variants) and of the many polarization metrics available, deriving an invariant relation among them. Secondly, we derive the conditions under which the FJ variants are able to induce opinion polarization in a social network, as a function of the social ties between the nodes and their individual susceptibility to the opinion of others. Thirdly, we discuss a methodology for finding concrete opinion vectors that are able to bring the network to a polarized state. Finally, our analytical results are applied to two real social network graphs, showing how our theoretical findings can be used to identify polarizing conditions under various configurations.