Fur a linear discrete-time system (C, A, B), the case when its transition matrixAis singular is studied, and it is shown that it can be decomposed into a parallel connection of two subsystems which are respectively nilpotent and regular. Using this result. it is shown that it is controllable if and only if rank [z/n- A B]=nfor any nonzero complex 2. The same result is derived for constructibility.