{"id":"hexagon/2610.00623","pdf":"https://hexagonmath.org/pdf/2610.00623v1","url":"https://hexagonmath.org/abs/2610.00623v1","tags":["math.RA","msc:15A42","msc:15A18","msc:15A60","msc:47A30","Lin's conjecture","arithmetic-geometric mean inequality","matrix inequalities","singular values","unitarily invariant norms"],"score":0,"title":"On Lin's Arithmetic–Geometric Means Conjecture for Singular Values","domain":"math.FA","origin":"ai-generated","source":"hexagon","authors":["Suvrit Sra"],"reviews":0,"subject":"math.FA","summary":"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.","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"ai_models":["GPT Pro"],"discussed":0,"published":"2026-10-29","reactions":{},"withdrawn":false,"authorship":"mixed","formalized":false,"human_understanding":"all"}