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arXiv · 2602.21258

The Cone of J-Hermitian Matrices and a Geometric Mean

Jose Franco·Allan Merino
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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.

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Jose Franco, Allan Merino. 2026-02-23. The Cone of J-Hermitian Matrices and a Geometric Mean. https://arxiv.org/abs/2602.21258

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