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On Lin's Arithmetic–Geometric Means Conjecture for Singular Values

A mix of human-written and AI-generated textHuman understanding: all partsmath.FA — Functional Analysismath.RA — Rings and Algebras

Contributed by Suvrit Sra ↗

Version 1 / Oct 29, 2026 / CC BY 4.0

Abstract

Let AA and BB be positive semidefinite, and let 0≤t≤10\le t\le 1. Lin (2017) conjectured that λj((1−t)A+tB)≥σj1/2(A2(1−t)B2t)\lambda_j((1-t)A+tB)\ge\sigma_j^{1/2}(A^{2(1-t)}B^{2t}) for j=1,…,nj=1,\ldots,n. The case t=1/2t=1/2 is Drury's theorem, which answered a long-standing question of Bhatia and Kittaneh. Lin remarked that Drury's method does not seem to reach other values of tt. We prove his conjecture for every t∈[0,1]t\in[0,1]. The proof builds on Drury's argument, but replaces his block-matrix step (unique to the midpoint t=1/2t=1/2) by an eigenvalue count based on an inertia identity and a one-variable inequality.

Provenance statement

The results in this paper were developed with an AI assistant (OpenAI's GPT Pro). None of the claims were obtained zero-shot: they emerged through sustained back-and-forth discussion, including failed attempts, counterexamples, audits, and repairs.

Tools used

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GPT Pro

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Version history

  1. v1Submitted by Suvrit SraInitial depositCurrentOct 28, 2026