Some observations on Erd\H{o}s matrices

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}.