Kalman’s or Hautus’ controllability (observability) criteria may reveal that a system is not completely state controllable (observable), which does not mean that the system is not controllable (observable) and suggests to decompose the system into subsystems. The chapter presents the transformation into the controllable normal form (observable normal form) and Kalman’s general decomposition of a state-space model that is neither completely state controllable nor completely state observable. The linear subspace that is completely state controllable and completely state observable is of particular interest. Its matrices build a minimal state-space representation of a transfer function matrix relating inputs and outputs. For uncontrollable (unobservable) systems, it is important to check whether they are stabilisable (detectable). This checks also the transformation to controllable normal form (observable normal form).