Log-majorizations for the (symplectic) eigenvalues of the Cartan barycenter

TL;DR

This paper demonstrates Lipschitz continuity of eigenvalue maps for positive definite matrices under the Cartan-Hadamard metric and establishes log-majorizations for eigenvalues of the Cartan barycenter, leading to a Jensen's inequality for matrix-valued integrals.

Contribution

It introduces log-majorization results for (symplectic) eigenvalues of the Cartan barycenter, extending matrix inequalities to a probabilistic and geometric setting.

Findings

Eigenvalue map is Lipschitz continuous under the Cartan-Hadamard metric.

Log-majorizations for (symplectic) eigenvalues of the Cartan barycenter are established.

A Jensen's inequality for geometric integrals of matrix-valued random variables is derived.

Abstract

In this paper we show that the eigenvalue map and the symplectic eigenvalue map of positive definite matrices are Lipschitz for the Cartan-Hadamard Riemannian metric, and establish log-majorizations for the (symplectic) eigenvalues of the Cartan barycenter of integrable probability Borel measures. This leads a version of Jensen's inequality for geometric integrals of matrix-valued integrable random variables.