The paper considers the dynamics of discrete-time polytopic systems. Two types of results are presented. The first type refers to the estimation of the spectrum location for the matrix polytope that defines the behavior of a given system. The approach relies on the theory of representatives applied to a set of nonnegative matrices built from the vertices of the matrix polytope. The estimation is provided as an outer bound for the spectrum location, which is expressed in terms of the dominant eigenvalues of the row and column representatives associated with the aforementioned set of nonnegative matrices. The second type of results discusses the dynamics of the polytopic systems by analyzing the existence of Lyapunov functions with diagonal forms. It is shown that whenever such Lyapunov functions exist, their decreasing rate is related to the concrete values of the dominant eigenvalues of the row and column representatives. The applicability of the theoretical developments is illustrated by two case studies. The first one is a numerical example, whereas the second proves that some results previously reported in literature can be incorporated as particularizations of the current (more general) approach.