Generic Features in the Spectral Decomposition of Correlation Matrices

TL;DR

This paper investigates the spectral properties of correlation matrices, establishing bounds on eigenvalues and eigenvector alignment, and explaining the conditions under which large eigenvalues and diagonal eigenvectors naturally occur.

Contribution

It introduces geometric methods to derive bounds on spectral features of correlation matrices and explains their generic occurrence independent of matrix size.

Findings

Lower bounds for the largest eigenvalue derived

Conditions for eigenvector alignment explained

Large eigenvalues occur independently of matrix dimension

Abstract

We show that correlation matrices with particular average and variance of the correlation coefficients have a notably restricted spectral structure. Applying geometric methods, we derive lower bounds for the largest eigenvalue and the alignment of the corresponding eigenvector. We explain how and to which extent, a distinctly large eigenvalue and an approximately diagonal eigenvector generically occur for specific correlation matrices independently of the correlation matrix dimension.