Entrywise nonnegativity of the inverse relative gain array in dimension six
Abstract
We prove that (G∘ G−1)−1 is entrywise nonnegative for every real symmetric positive definite matrix G of order at most six. Its entries are strictly positive exactly within the connected components of the nonzero-entry graph of G. An explicit rational example of order seven establishes sharpness. We give two complementary structural proofs. The analytic proof combines a trace obstruction with continuation of positive column forms and a bordering identity. The algebraic proof establishes a stronger four-dimensional polarized inequality through an explicit ten-square polynomial certificate. A common framework relates biquadratic positivity, column forms, positive semidefinite precision loads, and inverse relative gain array nonnegativity in consecutive dimensions. Further consequences include exact load-deficit identities, polynomial sum-of-squares certificates with the covariance fixed, and a reverse Schur--Horn majorization inequality with the same sharp dimension bound. In every dimension, sufficiently large powers of the inverse relative gain array are doubly stochastic.
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BibTeX
@misc{uhlmann2026entrywise,
title = {{Entrywise nonnegativity of the inverse relative gain array in dimension six}},
author = {Jeffrey Uhlmann and Jie Wang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.35558},
eprint = {2609.35558},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.35558},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Entrywise nonnegativity of the inverse relative gain array in dimension six
AU - Jeffrey Uhlmann
AU - Jie Wang
PY - 2026
DA - 2026/10/21
PB - arXiv
DO - 10.48550/arxiv.2609.35558
UR - https://arxiv.org/abs/2609.35558
AB - We prove that (G∘ G^−1)^−1 is entrywise nonnegative for every real symmetric positive definite matrix G of order at most six. Its entries are strictly positive exactly within the connected components of the nonzero-entry graph of G. An explicit rational example of order seven establishes sharpness. We give two complementary structural proofs. The analytic proof combines a trace obstruction with continuation of positive column forms and a bordering identity. The algebraic proof establishes a stronger four-dimensional polarized inequality through an explicit ten-square polynomial certificate. A common framework relates biquadratic positivity, column forms, positive semidefinite precision loads, and inverse relative gain array nonnegativity in consecutive dimensions. Further consequences include exact load-deficit identities, polynomial sum-of-squares certificates with the covariance fixed, and a reverse Schur--Horn majorization inequality with the same sharp dimension bound. In every dimension, sufficiently large powers of the inverse relative gain array are doubly stochastic.
ER -