发表机构
State Key Laboratory of Mathematical Sciences, Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院数学科学重点实验室）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明了六阶及以下实对称正定矩阵的逆相对增益阵列逐项非负，解决猜想，并推广到维度间的等价性及反向优超不等式。
AI 中文摘要
我们证明了对于阶数至多为六的每个实对称正定矩阵 $G$，$(G\circ G^{-1})^{-1}$ 是逐项非负的，从而解决了 Jeffrey Uhlmann 的逆相对增益阵列猜想。该证明结合了 $\Lambda^2\mathbb R^4$ 上的显式平方和恒等式与两个加边论证。更一般地，我们建立了双二次不等式、二次正定性和三个连续维度中逆相对增益阵列非负性之间的等价性。作为推论，我们刻画了逆相互作用算子的固定空间和平均性质，并证明了由实对称正定矩阵对角化的矩阵的一个反向 Schur--Horn 优超不等式。一个七阶有理示例表明两个维度界限都是紧的。
英文摘要
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.
Comments14 pages
