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| Natura: | Preprint |
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2026
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| Accesso online: | https://arxiv.org/abs/2602.21258 |
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| _version_ | 1866917293332627456 |
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| author | Franco, Jose Merino, Allan |
| author_facet | Franco, Jose Merino, Allan |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_21258 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The Cone of J-Hermitian Matrices and a Geometric Mean Franco, Jose Merino, Allan Differential Geometry Functional Analysis Operator Algebras Primary: 15A42, Secondary: 47A63 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. |
| title | The Cone of J-Hermitian Matrices and a Geometric Mean |
| topic | Differential Geometry Functional Analysis Operator Algebras Primary: 15A42, Secondary: 47A63 |
| url | https://arxiv.org/abs/2602.21258 |