Linear Systems as Representations of Time Groups

Abstract

In this paper, we develop a representation-theoretic formulation of discrete-time linear systems. We show that such systems are naturally viewed as representations of time groups acting on vector spaces, thereby endowing the state space with a canonical algebraic structure. This perspective provides a unified framework for linear systems over different fields, in which familiar structural properties arise from the underlying representation. In particular, invariant decompositions of the state space correspond to invariant subrepresentations, while the distinctions between real, complex, and finite-field systems emerge from the algebraic properties of the base field and the time group. We further show that linear systems over finite fields naturally correspond to representations of finite cyclic time groups, leading to module structures over polynomial quotient rings. This provides a systematic alternative to spectral analysis in settings where eigenvalue-based methods are not the most natural organizing language.