Some observations on Erdős matrices

Abstract

In a seminal paper in 1959, Marcus and Ree proved that every $n\times n$ bistochastic matrix A satisfies ${\Vert A\Vert }_F^2\le {\max }_{\sigma \in S_n}{A}_{i,\sigma \left(i\right)}$ where $S_n$ is the symmetric group on $\{1,\dots ,n\}$. Erdős asked to characterize the bistochastic matrices for which the equality holds in the Marcus–Ree inequality. We refer to such matrices as Erdős matrices. While this problem is trivial in dimension $n=2$, the case of dimension $n=3$ was only resolved recently in [4] in 2023. We prove that for every n, there are only finitely many $n\times n$ Erdős matrices. We also give a complete characterization of Erdős matrices that yields an algorithm to generate all Erdős matrices in any given dimension. We also prove that Erdős matrices can have only rational entries. This answers a question of [4].