An ensemble of high rank matrices arising from tournaments
Fuente:
arXiv
Enregistré dans:
| Auteurs principaux: | , , |
|---|---|
| Format: | Preprint |
| Publié: |
2021
|
| Sujets: | |
| Accès en ligne: | |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866911067222835200 |
|---|---|
| author | Balachandran, Niranjan Bhattacharya, Srimanta Sankarnarayanan, Brahadeesh |
| author_facet | Balachandran, Niranjan Bhattacharya, Srimanta Sankarnarayanan, Brahadeesh |
| contents | Suppose $\mathbb{F}$ is a field and let $\mathbf{a} := (a_1, a_2, \dotsc)$ be a sequence of non-zero elements in $\mathbb{F}$. For $\mathbf{a}_n := (a_1, \dotsc, a_n)$, we consider the family $\mathcal{M}_n(\mathbf{a})$ of $n \times n$ symmetric matrices $M$ over $\mathbb{F}$ with all diagonal entries zero and the $(i, j)$th element of $M$ either $a_i$ or $a_j$ for $i < j$. In this short paper, we show that all matrices in a certain subclass of $\mathcal{M}_n(\mathbf{a})$ -- which can be naturally associated with transitive tournaments -- have rank at least $\lfloor 2n/3 \rfloor - 1$. We also show that if $\operatorname{char}(\mathbb{F}) \neq 2$ and $M$ is a matrix chosen uniformly at random from $\mathcal{M}_n(\mathbf{a})$, then with high probability $\operatorname{rank}(M) \geq \bigl(\frac{1}{2} - o(1)\bigr)n$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2108_10871 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | An ensemble of high rank matrices arising from tournaments Balachandran, Niranjan Bhattacharya, Srimanta Sankarnarayanan, Brahadeesh Combinatorics 15B52 (Primary) 05D99, 05C20, 15A03 (Secondary) Suppose $\mathbb{F}$ is a field and let $\mathbf{a} := (a_1, a_2, \dotsc)$ be a sequence of non-zero elements in $\mathbb{F}$. For $\mathbf{a}_n := (a_1, \dotsc, a_n)$, we consider the family $\mathcal{M}_n(\mathbf{a})$ of $n \times n$ symmetric matrices $M$ over $\mathbb{F}$ with all diagonal entries zero and the $(i, j)$th element of $M$ either $a_i$ or $a_j$ for $i < j$. In this short paper, we show that all matrices in a certain subclass of $\mathcal{M}_n(\mathbf{a})$ -- which can be naturally associated with transitive tournaments -- have rank at least $\lfloor 2n/3 \rfloor - 1$. We also show that if $\operatorname{char}(\mathbb{F}) \neq 2$ and $M$ is a matrix chosen uniformly at random from $\mathcal{M}_n(\mathbf{a})$, then with high probability $\operatorname{rank}(M) \geq \bigl(\frac{1}{2} - o(1)\bigr)n$. |
| title | An ensemble of high rank matrices arising from tournaments |
| topic | Combinatorics 15B52 (Primary) 05D99, 05C20, 15A03 (Secondary) |
| url | https://arxiv.org/abs/2108.10871 |