Intermediate and Limiting Behavior of Powers of Some Circulant Matrices

Abstract

Let $A$ be an arbitrary circulant stochastic matrix, and let ${\underset{\_}{x}}_0$ be a vector. An “asymptotic” canonical form is derived for $A^k$ (as $k\to \mathrm{\infty }$) as a tensor product of three simple matrices by employing a pseudo-invariant on sections of states for a Markov process with transition matrix $A$, and by analyzing how $A$ acts on the sections, through its auxiliary polynomial. An element-wise asymptotic characterization of $A^k$ is also given, generalizing previous results to cover both periodic and aperiodic cases. For a particular circulant stochastic matrix, identifying the intermediate stage at which fractions first appear in the sequence ${\underset{\_}{x}}_k=A^k{\underset{\_}{x}}_0$, is accomplished by utilizing congruential matrix identities and $(0,1)$-matrices to determine the minimum $2$-adic order of the coordinates of ${\underset{\_}{x}}_k$, through their binary expansions. Throughout, results are interpreted in the context of an arbitrary weighted average repeatedly applied simultaneously to each term of a finite sequence when read cyclically.