On the eigenvalues of the spatial sign covariance matrix in more than two dimensions

TL;DR

This paper investigates the eigenvalues of the spatial sign covariance matrix for elliptical distributions, revealing their relationship with the shape matrix eigenvalues and providing a numerical computation method.

Contribution

It establishes a one-to-one correspondence between the eigenvalues of the spatial sign covariance matrix and the shape matrix, and introduces an integral representation for easier computation.

Findings

Eigenvalues are uniquely determined by the shape matrix eigenvalues.

Eigenvalues of the spatial sign covariance matrix are more tightly clustered.

Provides a practical integral formula for numerical evaluation.

Abstract

We gather several results on the eigenvalues of the spatial sign covariance matrix of an elliptical distribution. It is shown that the eigenvalues are a one-to-one function of the eigenvalues of the shape matrix and that they are closer together than the latter. We further provide a one-dimensional integral representation of the eigenvalues, which facilitates their numerical computation.