It is proved that the system matrix P(s) of anl-input,m-output controllable system 𝒮 can be brought by strict system equivalence transformations into a special form [Ptilde](s) such that the system 𝒮 is described by the module μ generated by the firstLcolumn vectors of the matrix [Ptilde](s). Invariants of this module are determined and related to invariants of the system 𝒮 under similarity transformations, such as the controllability indices and the controllability subspaces.