Partial Proof of a Conjecture with Implications for Spectral Majorization
This work addresses a theoretical problem in linear algebra with potential implications for matrix theory, but it is incremental as it builds on existing methods and focuses on specific matrix sizes.
The paper tackles a conjecture about spectral majorization for positive definite matrices up to size 6, proving it for n≤4 using computer-assisted sum of squares methods and identifying a new family of matrices with this property. It extends this family to larger matrices via Kronecker composition, retaining the majorization property.
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.