The Cone of J-Hermitian Matrices and a Geometric Mean

TL;DR

This paper explores the geometric and algebraic structure of the cone of positive J-Hermitian matrices, introducing a J-geometric mean via geodesics and establishing its unique characterization through Riccati equations.

Contribution

It introduces a novel geometric mean for J-Hermitian matrices and analyzes the structure of the associated cone using the J-exponential map.

Findings

J-exponential map is bijective on the cone

The cone admits a natural Riemannian structure

J-geometric mean is uniquely characterized as a Riccati equation solution

Abstract

We study the cone $\mathscr{P}_{\text{J}}$ of positive J-Hermitian matrices associated with an indefinite signature matrix J = $\text{Id}_{p,q}$. We show that the J-exponential map is bijective and use it to analyze the algebraic and geometric structure of $\mathscr{P}_{\text{J}}$. Through a canonical identification with the cone of positive definite matrices, we endow $\mathscr{P}_{\text{J}}$ with a natural Riemannian structure. In this setting, we define a J-geometric mean as the midpoint of geodesics and prove that it is uniquely characterized as the solution of a Riccati-type equation.