Partial Proof of a Conjecture with Implications for Spectral
Majorization
- URL: http://arxiv.org/abs/2309.01302v1
- Date: Mon, 4 Sep 2023 01:02:19 GMT
- Title: Partial Proof of a Conjecture with Implications for Spectral
Majorization
- Abstract summary: We present new results relating to a conjecture regarding properties of $ntimes n$, $nleq 6$, positive definite matrices.
We conclude with general considerations on the future of computer-assisted and AI-based proofs.
- Score: 0.43512163406551996
- License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
- 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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