Some observations on Erd\H{o}s matrices

TL;DR

This paper investigates Erdős matrices, which are bistochastic matrices satisfying a specific equality, proving finiteness in each dimension, providing a characterization and algorithm for their generation, and showing they have only rational entries.

Contribution

It establishes the finiteness of Erdős matrices in each dimension, offers a characterization and an algorithm to generate all such matrices, and proves they have only rational entries.

Findings

Only finitely many Erdős matrices exist in each dimension.

An explicit characterization and generation algorithm for Erdős matrices.

Erdős matrices can only have rational entries.

Abstract

In a seminal paper in 1959, Marcus and Ree proved that every $n\times n$ bistochastic matrix $A$ satisfies $\|A\|_{\operatorname{F}}^2\leq \max_{\sigma\in S_n}A_{i,\sigma(i)}$ where $S_n$ is the symmetric group on $\{1, \ldots, n\}$. Erd\H{o}s asked to characterize the bistochastic matrices for which the equality holds in the Marcus--Ree inequality. We refer to such matrices as Erd\H{o}s matrices. While this problem is trivial in dimension $n=2$, the case of dimension $n=3$ was only resolved recently in~\cite{bouthat2024question} in 2023. We prove that for every $n$, there are only finitely many $n\times n$ Erd\H{o}s matrices. We also give a characterization of Erd\H{o}s matrices that yields an algorithm to generate all Erd\H{o}s matrices in any given dimension. We also prove that Erd\H{o}s matrices can have only rational entries. This answers a question of~\cite{bouthat2024question}.