Some observations on Erd\H{o}s matrices
In a seminal paper in 1959, Marcus and Ree proved that every $n\times n$ bistochastic matrix $A$ satisfies $\|A\|_{F}^2\leq \max_{\sigma\in S_n}A_{i,\sigma(i)}$ 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...
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| Main Author | |
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| Format | Journal Article |
| Language | English |
| Published |
09.10.2024
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| Subjects | |
| Online Access | Get full text |
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| Summary: | In a seminal paper in 1959, Marcus and Ree proved that every $n\times n$
bistochastic matrix $A$ satisfies $\|A\|_{F}^2\leq \max_{\sigma\in
S_n}A_{i,\sigma(i)}$ 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 resolved 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}. |
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| DOI: | 10.48550/arxiv.2410.06612 |