  Discrete algebraic Riccati equations and their fixed points are well
understood and arise in a variety of applications, however, the time-varying
equations have not yet been fully explored in the literature. In this article
we provide a self-contained study of discrete time Riccati matrix difference
equations. In particular, we provide a novel Riccati semigroup duality formula
and a new Floquet-type representation for these equations. Due to the
aperiodicity of the underlying flow of the solution matrix, conventional
Floquet theory does not apply in this setting and thus further analysis is
required. We illustrate the impact of these formulae with an explicit
description of the solution of time-varying Riccati difference equations and
its fundamental-type solution in terms of the fixed point of the equation and
an invertible linear matrix map, as well as uniform upper and lower bounds on
the Riccati maps. These are the first results of this type for time varying
Riccati matrix difference equations.
