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The Cone of J-Hermitian Matrices and a Geometric Mean

Jose Franco, Allan Merino

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.

The Cone of J-Hermitian Matrices and a Geometric Mean

Abstract

We study the cone of positive J-Hermitian matrices associated with an indefinite signature matrix J = . We show that the J-exponential map is bijective and use it to analyze the algebraic and geometric structure of . Through a canonical identification with the cone of positive definite matrices, we endow 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.
Paper Structure (12 sections, 38 theorems, 249 equations)

This paper contains 12 sections, 38 theorems, 249 equations.

Table of Contents

  1. Introduction
  2. The cone of J-Hermitian matrices
  3. Square root on $\mathscr{P}_{{\rm J}}$ and properties
  4. Loewner order on the cone $\mathscr{P}_{{\rm J}}$
  5. Geodesics on the cone $\mathscr{P}_{{\rm J}}$
  6. Geometric mean on the cone $\mathscr{P}_{{\rm J}}$
  7. The quaternionic case
  8. The division algebra $\mathbb{H}$
  9. The cone of positive quaternionic hermitian matrices
  10. A Riemannian form on $\mathscr{P}$
  11. Geodesics on $\mathscr{P}$
  12. Geometric mean on $\mathscr{P}$

Key Result

Theorem 1.1

Let ${\rm A},{\rm B}\in \mathscr{P}_{{\rm J}}$. The map $\gamma: \left[0\,, 1\right] \to \mathscr{P}_{{\rm J}}$ given by is the equation of the geodesic between ${\rm A}$ and ${\rm B}$ in $\mathscr{P}_{{\rm J}}$ with respect to $\omega$, where ${\rm X}^{t}_{{\rm J}}$ is the matrix in $\mathscr{P}_{{\rm J}}$ given by

Theorems & Definitions (104)

  • Theorem 1.1
  • Remark 2.1
  • Theorem 2.3: Sylvester's law of inertia
  • proof
  • Remark 2.4
  • Remark 2.5
  • Definition 2.6
  • Remark 2.8
  • Lemma 2.9
  • proof
  • ...and 94 more