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A Spectral Blakley–Roy Inequality for Nonnegative Matrices

A mix of human-written and AI-generated textHuman understanding: all partsmath.CO — Combinatoricsmath.SP — Spectral Theorymath.RA — Rings and Algebras

Contributed by Suvrit Sra ↗

Version 1 / Oct 29, 2026 / CC BY 4.0

Abstract

Let AA and BB be real symmetric matrices with nonnegative entries. We prove that ρ(AB)k≤ρ(AkBk)\rho(AB)^k\le\rho(A^kB^k) for the spectral radius ρ(⋅)\rho(\cdot) and every integer k≥1k\ge1; for B=uuTB=uu^T this is the Blakley–Roy inequality. We prove it via the more general inequality ∥(AB)m∥k≤∥(AkBk)m∥\|(AB)^m\|^{k}\le\|(A^kB^k)^m\| in the spectral norm, valid for all integers k,m≥1k,m\ge1. The even exponents case follows from standard tools; for odd exponents k=2r+1k=2r+1 the claim follows via the more general product inequality (uTC1⋯Ctv)k≤(uTu)r(vTv)r uTC1k⋯Ctkv(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 CiC_i and nonnegative u,vu,v. Its proof is an entropy argument on paths that traverse each edge kk times, back and forth.

Provenance statement

The results in this paper were developed through extensive use of AI assistants: GPT Pro for the main line of argument, and Claude for the writing, literature search, and some examples. None of the claims were obtained zero-shot: they emerged through sustained back-and-forth discussion, including failed attempts, counterexamples, audits, and repairs. The main idea to pursue the norm inequality was introduced by the author.

Tools used

OpenAI
GPT Pro
Anthropic
Claude

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Version history

  1. v1Submitted by Suvrit SraInitial depositCurrentOct 28, 2026