Research Preprints · Matrix analysis
A Spectral Blakley–Roy Inequality for Nonnegative Matrices
Abstract
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.
Versions
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 | 15 October 2026 | 10 pp | .ots | — |
Cite
@misc{sra2026pourya-rho-v1,
author = {Suvrit Sra},
title = {{A Spectral Blakley--Roy Inequality for Nonnegative Matrices}},
year = {2026},
month = oct,
howpublished = {Preprint, \url{https://github.com/suvrit/preprints/tree/main/preprints/pourya-rho/v1}},
note = {Version v1, released 15 October 2026},
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pip install opentimestamps-client curl -O https://suvrit.github.io/papers/pourya-rho/pourya-rho-v1.pdf -O https://suvrit.github.io/papers/pourya-rho/pourya-rho-v1.pdf.ots ots verify pourya-rho-v1.pdf.ots