When is a periodic discrete-time system equivalent to a time invariant one?

We give the precise conditions under which a periodic discrete-time linear state-space system can be transformed into a time-invariant one by a change of basis. Thus our theory is the discrete-time counterpart of the classical theory of Floquet transforms developed by Floquet and Lyapunov in the 1800s for continuous-time systems. We state and prove a necessary and sufficient condition for a “discrete-time Floquet transform” to exist, and give a construction for the transform when it does exist. Our results also extend to generalized state-space, or descriptor, systems