A Note On Convexity Inequalities Of Weighted Matrix Geometric Means

TL;DR

This paper provides a new proof of convexity inequalities for the space of positive-semidefinite matrices under the weighted matrix geometric mean, characterizing equality cases via matrix commutativity.

Contribution

It introduces a novel proof technique using log majorization for convexity inequalities in matrix geometric means, clarifying when equality holds.

Findings

Equality in convexity inequalities occurs if and only if matrices commute.

The proof leverages log majorization to characterize equality cases.

The results deepen understanding of geometric mean inequalities in matrix analysis.

Abstract

We offer a new proof of uniform convexity inequalities for the Finsler manifold of nonpositive curvature taken on the space of positive-semidefinite matrices with the weighted matrix geometric mean defining the geodesic between two points. Using the technique of log majorization, we are able to characterize that the equality cases of said equalities occur if and only if the matrices commute, and hence are the same as in $\ell^p$.