The Maximum Diagonal Product of a Stochastic Matrix of Bounded Rank

Let A = (aij) be a nonnegative row-stochastic or column-stochastic n × n matrix with rank A ≤ k, where 1 ≤ k ≤ n. Write n = qk + s, where 0 ≤ s < k. The author conjectured in 2018 that, for every permutation σ of {1, …, n}, ∏i=1n ai,σ(i) ≤ q−q(k−s)(q+1)−(q+1)s. We prove this bound and classify all equality cases.

Authors

Publication Details

Journal
Zenodo (CERN European Organization for Nuclear Research)
Published
2026-10-21
DOI
https://doi.org/10.5281/zenodo.23518363
Primary Topic
Matrix Theory and Algorithms
Type
preprint
Controls
|||
ALL TIME
JAN
FEB
MAR
APR
MAY
JUN
JUL
AUG
SEP
OCT
preprint

The Maximum Diagonal Product of a Stochastic Matrix of Bounded Rank

Yair Lavi
Zenodo (CERN European Organization for Nuclear Research)
Matrix Theory and Algorithms
preprint

The Maximum Diagonal Product of a Stochastic Matrix of Bounded Rank

Yair Lavi
preprint en

Abstract

Let A = (aij) be a nonnegative row-stochastic or column-stochastic n × n matrix with rank A ≤ k, where 1 ≤ k ≤ n. Write n = qk + s, where 0 ≤ s < k. The author conjectured in 2018 that, for every permutation σ of {1, …, n}, ∏i=1n ai,σ(i) ≤ q−q(k−s)(q+1)−(q+1)s. We prove this bound and classify all equality cases.

Zenodo (CERN European Organization for Nuclear Research)
Matrix Theory and Algorithms
DOIOpen Access PDFOpenAlex
AI Navigator

Ask Laika to Summarize, Analyze, and Connect papers live on the map.

Summarize Papers & Methodologies

Extract key findings, datasets, and comparative methods across publications.

Benchmark Rankings & Visual Analytics

Rank top research institutions, authors, funders, topics, and journals by Field-Weighted Citation Impact (FWCI) and paper volume with instant charts.

Connect Distant Disciplines

Bridge topological clusters on the map to find hidden collaborative intersections.

The Maximum Diagonal Product of a Stochastic Matrix of Bounded Rank — Yair Lavi · Zenodo (CERN European Organization for Nuclear Research) (2026) | TGRS Research Map | TGRS