Measuring Sphericity in Positive Semi-Definite Matrices

Abstract

The measure of sphericity for positive semi-definite matrices plays a crucial role in understanding their geometric properties, especially in high-dimensional settings. This paper introduces a robust measure of sphericity, which remains invariant under orthogonal transformations and scaling. We explore its behavior in finite-dimensional cases. Additionally, we investigate the stochastic case by considering a normal distribution, analyzing the asymptotic normality of random matrices and its implications on the convergence properties of the proposed measure.