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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 →

hexagon:2610.00602v1

math.FA — Functional Analysis

Tsallis Metrics on Density Matrices via the S-Divergence

Contributed by Suvrit Sra

We study symmetrized divergences induced by centroids of $\alpha$ -Tsallis relative entropies between density matrices $\sigma$ and $\rho$ ; up to scaling, the induced divergence equals $|2-\|\rho^\alpha+\sigma^\alpha\|_{1/\alpha}|$ . Earlier work (S. Sra, "Metrics induced by Jensen–Shannon and related divergences on positive definite matrices," Linear Algebra Appl., 2021) studied such divergences and metrics induced by them. That paper allowed centroids to range over the positive definite cone, while we require that the centroid itself be a density matrix. This (more natural) choice makes the metricity in this paper considerably stronger than that of the aforementioned paper. Specifically, we prove that $|2-\|\rho^\alpha+\sigma^\alpha\|_{1/\alpha}|^{1/2}$ is a metric on density matrices for $1/2\le\alpha\le2$ , $\alpha\ne1$ . For $\alpha<1$ , the proof uses a new four-point theorem for the S-divergence that might be of independent interest. For $\alpha>1$ , we first prove the endpoint $\alpha=2$ , and then transfer it to the other orders by a tensoring trick.

A mix of human-written and AI-generated textHuman understanding: all partsS-divergenceSchatten normSibson information radiusTsallis relative entropylog-determinant inequalitynegative typequantum Jensen-Shannon divergencequantum Renyi entropy

Oct 29, 2026PDF ↗View work →

Matching works

A Spectral Blakley–Roy Inequality for Nonnegative Matrices

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

Tsallis Metrics on Density Matrices via the S-Divergence

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