Clustering behaviour in Markov chains with eigenvalues close to one

Finite, discrete, time-homogeneous Markov chains are frequently used as a simple mathematical model of real-world dynamical systems. In many such applications, an analysis of clustering behaviour in the states of the system is desirable, and it is well-known that the eigendecomposition of the transition matrix A of the Markov chain can provide such insight. Clustering methods based on the sign pattern in the second eigenvector of A are frequently used when A has dominant eigenvalues that are real. In this paper, we present a method to include an analysis for complex eigenvalues of A which are close to 1. Since a real spectrum is not guaranteed in most applications, this is a valuable result in the area of spectral clustering in Markov chains.