A note on switching eigenvalues under small perturbations

TL;DR

This paper investigates how small perturbations can cause eigenvalues of symmetric matrices to switch ranks, affecting the interpretation of principal components and influence diagnostics, and proposes methods to detect such switching.

Contribution

It introduces a way to approximate eigenvalues to identify when eigenvalue switching occurs due to data perturbations, improving analysis robustness.

Findings

Eigenvalue switching can be detected using eigenvalue approximations.

Switching impacts the choice of principal components and influence diagnostics.

The approach is applicable to any symmetric matrix eigenvalue problem.

Abstract

Sensitivity of eigenvectors and eigenvalues of symmetric matrix estimates to the removal of a single observation have been well documented in the literature. However, a complicating factor can exist in that the rank of the eigenvalues may change due to the removal of an observation, and with that so too does the perceived importance of the corresponding eigenvector. We refer to this problem as "switching of eigenvalues". Since there is not enough information in the new eigenvalues post observation removal to indicate that this has happened, how do we know that this switching has occurred? In this paper, we show that approximations to the eigenvalues can be used to help determine when switching may have occurred. We then discuss possible actions researchers can take based on this knowledge, for example making better choices when it comes to deciding how many principal components should be retained and adjustments to approximate influence diagnostics that perform poorly when switching has occurred. Our results are easily applied to any eigenvalue problem involving symmetric matrix estimators. We highlight our approach with application to a real data example.