Entrywise nonnegativity of the inverse relative gain array in dimension six

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, resolving the inverse relative gain array conjecture of Jeffrey Uhlmann. The proof combines an explicit sum-of-squares identity on $Λ^2\mathbb R^4$ 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.