Volumen 15, Número 2, 2023

Partial Proof of a Conjecture with Implications for Spectral Majorization

Jeffrey Uhlmann


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.

DOI: https://doi.org/10.46571/JCI.2023.2.4

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