Year: 2026
Type: preprint
Abstract: We prove that $(G\circ G^{-1})^{-1}$ 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.
Source: arXiv (Cornell University)
Authors: Uhlmann, Jeffrey, Wang, Jie
Language: English
Topic: Matrix Theory and Algorithms
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
Awards: G3356097721, G7864953252
OpenAlex ID: W7214710616