Computing asymptotic eigenvectors and eigenvalues of perturbed symmetric matrices

TL;DR

This paper presents a simplified method for computing the asymptotic eigenvectors and eigenvalues of perturbed symmetric matrices, especially when classical approaches are complex, with applications in statistics and approximation theory.

Contribution

It introduces a new approach using successive Schur complements and generalized kernel forms to analyze eigenstructure in perturbed symmetric matrices, confirming a conjecture relating eigenvectors to orthogonal polynomials.

Findings

Derived explicit formulas for eigenvalues and eigenvectors in the flat limit.

Validated the approach by proving a conjecture linking eigenvectors to orthogonal polynomials.

Simplified classical complex integral expressions for eigenstructure analysis.

Abstract

Computing the eigenvectors and eigenvalues of a perturbed matrix can be remarkably difficult when the unperturbed matrix has repeated eigenvalues. In this work we show how the limiting eigenvectors and eigenvalues of a symmetric matrix $K(\varepsilon)$ as $\varepsilon \to 0$ can be obtained relatively easily from successive Schur complements, provided that the entries scale in different orders of $\varepsilon$. If the matrix does not directly exhibit this structure, we show that putting the matrix into a ``generalised kernel form'' can be very informative. The resulting formulas are much simpler than classical expressions obtained from complex integrals involving the resolvent. We apply our results to the problem of computing the eigenvalues and eigenvectors of kernel matrices in the ``flat limit'', a problem that appears in many applications in statistics and approximation theory. In particular, we prove a conjecture from [SIAM J. Matrix Anal. Appl., 2021, 42(1):17--57] which connects the eigenvectors of kernel matrices to multivariate orthogonal polynomials.