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math.CO — Combinatorics

A Spectral Blakley–Roy Inequality for Nonnegative Matrices

Contributed by Suvrit Sra

Let $A$ and $B$ be real symmetric matrices with nonnegative entries. We prove that $\rho(AB)^k\le\rho(A^kB^k)$ for the spectral radius $\rho(\cdot)$ and every integer $k\ge1$ ; for $B=uu^T$ this is the Blakley–Roy inequality. We prove it via the more general inequality $\|(AB)^m\|^{k}\le\|(A^kB^k)^m\|$ in the spectral norm, valid for all integers $k,m\ge1$ . The even exponents case follows from standard tools; for odd exponents $k=2r+1$ the claim follows via the more general product inequality $(u^TC_1\cdots C_tv)^k\le(u^Tu)^r(v^Tv)^r\,u^TC_1^k\cdots C_t^kv$ for nonnegative symmetric $C_i$ and nonnegative $u,v$ . Its proof is an entropy argument on paths that traverse each edge $k$ times, back and forth.

A mix of human-written and AI-generated textHuman understanding: all partsBlakley-Roy inequalitymatrix productsnonnegative matricesspectral radius

Oct 29, 2026PDF ↗View work →

hexagon:2610.00623v1

math.FA — Functional Analysis

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

Contributed by Suvrit Sra

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.

A mix of human-written and AI-generated textHuman understanding: all partsLin's conjecturearithmetic-geometric mean inequalitymatrix inequalitiessingular valuesunitarily invariant norms

Oct 29, 2026PDF ↗View work →

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A Spectral Blakley–Roy Inequality for Nonnegative Matrices

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

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