# Partial Proof of a Conjecture with Implications for Spectral Majorization

arXiv: https://arxiv.org/abs/2309.01302 · HTML: https://mlsanity.com/paper/2309.01302/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
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- First posted on arXiv 4 Sep 2023.

## Abstract (arXiv)

In this paper we report on new results relating to a conjecture regarding properties of n× n, n≤ 6, positive definite matrices. The conjecture has been proven for n≤ 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.