Quantum channels are central objects of study in quantum information, of which entanglement breaking channels are an interesting subset. Entanglement is considered a fundamental resource and property of quantum mechanics that we can use as a driving force for new theories and to continue research in this area. We exhibit the connection between stochastic matrix theory and the iterative behaviour of entanglement breaking channels and relate the Jordan forms of the entanglement breaking channels and stochastic matrices. We build on this perspective to study the fixed point theory for such channels. We further consider the nullspace structure of entanglement breaking channels, in particular proving that every operator space of trace zero matrices is the nullspace of such a channel. We connect the nullspace of entanglement breaking channels with private subspaces and present examples and discuss connections between quantum privacy and trace zero matrices. Finally, we draw a connection between random unitary channels and entanglement breaking channels.