Research Preprints · Matrix analysis
On Lin's Arithmetic–Geometric Means Conjecture for Singular Values
Abstract
Let $A$ and $B$ be positive semidefinite, and let $0\le t\le 1$. Lin (2017) conjectured that $\lambda_j((1-t)A+tB)\ge\sigma_j^{1/2}(A^{2(1-t)}B^{2t})$ for $j=1,\ldots,n$. The case $t=1/2$ 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 $t$. We prove his conjecture for every $t\in[0,1]$. The proof builds on Drury's argument, but replaces his block-matrix step (unique to the midpoint $t=1/2$) by an eigenvalue count based on an inertia identity and a one-variable inequality.
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Version numbers are this site's own; arXiv numbering, where present, is separate. Every released file carries an OpenTimestamps proof made on its release day.
| Version | Released | Length | File | Proof | arXiv |
|---|---|---|---|---|---|
| v1 | 8 October 2026 | 5 pp | .ots | — |
Cite
@misc{sra2026linconj-v1,
author = {Suvrit Sra},
title = {{On Lin's Arithmetic--Geometric Means Conjecture for Singular Values}},
year = {2026},
month = oct,
howpublished = {Preprint, \url{https://github.com/suvrit/preprints/tree/main/preprints/linconj/v1}},
note = {Version v1, released 8 October 2026},
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pip install opentimestamps-client curl -O https://suvrit.github.io/papers/linconj/linconj-v1.pdf -O https://suvrit.github.io/papers/linconj/linconj-v1.pdf.ots ots verify linconj-v1.pdf.ots