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arXiv · 2309.01302

Partial Proof of a Conjecture with Implications for Spectral Majorization

Jeffrey Uhlmann
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Abstract

In this paper we report on new results relating to a conjecture regarding properties of $n\times n$, $n\leq 6$, positive definite matrices. The conjecture has been proven for $n\leq 4$ using computer-assisted sum of squares (SoS) methods for proving polynomial nonnegativity. Based on these proven cases, we report on the recent identification of a new family of matrices with the property that their diagonals majorize their spectrum. We then present new results showing that this family can extended via Kronecker composition to $n>6$ while retaining the special majorization property. We conclude with general considerations on the future of computer-assisted and AI-based proofs.

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Jeffrey Uhlmann. 2023-09-04. Partial Proof of a Conjecture with Implications for Spectral Majorization. https://arxiv.org/abs/2309.01302

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