On the irreducibility condition in the structural controllability theorem

It is known that the structural system (A,B) is structurally controllable if and only if the corresponding matrix [A B] is generically full rank and irreducible. In this paper it is shown that the irreducibility condition alone implies that every nonzero mode of (A,B) is generically controllable. This result provides an easy proof to the structural controllability theorem stated above. In addition, it is shown that the basic structure of the Jordan canonical form of (A,B) remains unaffected, in the generic sense, under the variation of the free parameters of (A,B).