Floquet transformations for discrete-time systems: Equivalence between periodic systems and time-invariant ones

This paper considers discrete-time periodic linear systems and their Floquet transformations, by which the periodic systems can be equivalently transformed to time-invariant ones. When the system matrix is nonsingular, it is very easy to derive the discrete-time Floquet transformation, however, when the system matrix is singular, the derivation is essentially different from the continuous-time case. A necessary and sufficient condition for Floquet transformation to exist has been already known and this paper aims to give a simpler proof, by which all the similarity classes of Floquet transformations could be made clear. And also an extension of Floquet transformations to non-autonomous systems is shown.