Fixed Points, Stability, Basin Geometry, and Global Convergence of the $3\times3$ Correlation Map

Abstract

For more than half a century, iterated Pearson correlation has underpinned methods for network blockmodeling, clustering, seriation, and information visualization. Despite its continued use and the longstanding assumption of convergence, the global convergence problem remained open even in dimension three. We resolve this problem completely for the $3\times3$ correlation map: every admissible orbit is well defined for all forward iterates and converges to one of exactly seven fixed points. In arbitrary dimension, we establish an exact row-wise Gram factorization and the identity $\operatorname{rank}C(A)=\operatorname{rank}(AH_n)$, which gives the precise rank-reduction mechanism and forces every fixed point to be singular. We identify the all-ones matrix as the unique Pearson-degenerate correlation matrix, prove forward invariance of the Pearson-nondegenerate elliptope, and classify the $2^{n-1}-1$ nondegenerate sign-valued fixed points. For $n=3$, we prove a complete analytical fixed-point classification: three rank-one patterned points, three rank-two mixed points, and one rank-two equicorrelation point. We also prove the complete relative Lyapunov stability classification: precisely the patterned points are locally asymptotically stable, while the mixed and equicorrelation points are unstable. The global proof reduces the rank-two dynamics to a one-dimensional projective kernel coordinate; an exact order identity produces monotone projective ratios, excludes nontrivial periodic and recurrent limit sets, and forces convergence to a fixed point. Finally, the initial conditions converging to the four unstable fixed points form a Lebesgue-null set. Hence almost every admissible initial condition converges to a patterned fixed point; the three patterned basins are relatively open and permutation-equivalent, and each has Lebesgue measure exactly one third of that of the elliptope.