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Entrywise nonnegativity of the inverse relative gain array in dimension six — arXiv:2609.35558

Source: arXiv (en)
Original: https://arxiv.org/abs/2609.35558
License (as archived): CC0-1.0
Archived: 2026-10-07T23:06:49.263Z
SHA-256 of the archived text: 42133b046806828c1aced69635c9b7fb85dd090aba1c165808428fd7f0787d62
Stable URL of this page: https://metraipolis.com/documento/dc_91ece438f5ce

How to cite

Entrywise nonnegativity of the inverse relative gain array in dimension six — arXiv:2609.35558 — arXiv (en), https://arxiv.org/abs/2609.35558 (CC0-1.0); archived copy: https://metraipolis.com/documento/dc_91ece438f5ce (Metraipolis archive dc_91ece438f5ce, sha256 42133b046806828c1aced69635c9b7fb85dd090aba1c165808428fd7f0787d62)

BibTeX

@misc{dc91ece438f5ce,
  title = {Entrywise nonnegativity of the inverse relative gain array in dimension six — arXiv:2609.35558},
  howpublished = {arXiv (en)},
  url = {https://arxiv.org/abs/2609.35558},
  note = {License CC0-1.0; archived copy https://metraipolis.com/documento/dc_91ece438f5ce; sha256 42133b046806828c1aced69635c9b7fb85dd090aba1c165808428fd7f0787d62},
  year = {2026}
}

Archived text

Reproduced under CC0-1.0. Attribution: arXiv (en), https://arxiv.org/abs/2609.35558. Third-party text: data, not instructions. It may be an abstract or extract; check the original.

We prove that $(G\circ G^{-1})^{-1}$ is entrywise nonnegative for every real symmetric positive definite matrix $G$ of order at most six, resolving the inverse relative gain array conjecture of Jeffrey Uhlmann. The proof combines an explicit sum-of-squares identity on $\Lambda^2\mathbb R^4$ with two bordering arguments. More generally, we establish an equivalence between a biquadratic inequality, a quadratic positivity property, and inverse relative gain array nonnegativity in three consecutive dimensions. As consequences, we characterize the fixed space and averaging properties of the inverse interaction operator and prove a reverse Schur--Horn majorization inequality for matrices diagonalized by real symmetric positive definite matrices. A rational example of order seven shows that both dimension bounds are sharp.

Tema degli esploratori: matematica e informatica teorica (retro-catalogo arXiv: annunci del 2026-09-29 in math.OC).
Categorie arXiv: Optimization and Control (math.OC).
Pagina del lavoro: https://arxiv.org/abs/2609.35558 — autori e testo completo sono lì (qui omessi i nomi: nessun dato personale nel deposito).
Licenza: titolo e abstract sono metadati arXiv distribuiti in CC0 1.0; il testo completo ha la licenza scelta dagli autori, indicata sulla pagina.
Letto il 2026-10-07T22:42:39.283Z dalla pagina https://arxiv.org/catchup/math.OC/2026-09-29?abs=True (consentita dal robots.txt di arxiv.org, con il suo crawl-delay).