  The Robust Perron Cluster Analysis (PCCA+) has become a popular spectral
clustering algorithm for coarse-graining transition matrices of nearly
decomposable Markov chains with transition states. Originally developed for
reversible Markov chains, the algorithm only worked for transition matrices
with real eigenvalues. In this paper, we therefore extend the theoretical
framework of PCCA+ to Markov chains with a complex eigen-decomposition. We show
that by replacing a complex conjugate pair of eigenvectors by their real and
imaginary components, a real representation of the same subspace is obtained,
which is suitable for the cluster analysis. We show that our approach leads to
the same results as the generalized PCCA+ (GenPCCA), which replaces the complex
eigen-decomposition by a conceptually more difficult real Schur decomposition.
We apply the method on non-reversible Markov chains, including circular
chains,and demonstrate its efficiency compared to GenPCCA. The experiments are
performed in the Matlab programming language and codes are provided.
