About subspaces the most deviating from the coordinate ones
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866913111792943104 |
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| author | Nesterenko, Yuri |
| author_facet | Nesterenko, Yuri |
| contents | Using the largest principal angle as a distance between same-dimensional linear subspaces of $\mathbb{R}^n$, we construct $k$-dimensional subspaces which deviate from every coordinate $k$-subspace by at least $\arccos(1/\sqrt n)$. The construction is motivated by the hypothesis of Goreinov, Tyrtyshnikov and Zamarashkin that this value is the largest possible one for all $n > k > 0$. The subspaces are scaled star spaces of $2$-connected series-parallel graphs with $k+1$ vertices and $n$ edges, equipped with a particular positive edge weighting, while the largest principal angles take two values -- $\arccos(1 / \sqrt{n})$ and $π/2$, depending on whether a $k$-edge subgraph corresponding to a coordinate $k$-subspace is a spanning tree or not.
For a fixed series-parallel graph, we also prove that the constructed weighting is the unique positive one, up to scaling, for which the corresponding $k$-subspace deviates from all coordinate $k$-subspaces by at least $\arccos(1 / \sqrt{n})$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_02387 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | About subspaces the most deviating from the coordinate ones Nesterenko, Yuri Numerical Analysis Combinatorics Using the largest principal angle as a distance between same-dimensional linear subspaces of $\mathbb{R}^n$, we construct $k$-dimensional subspaces which deviate from every coordinate $k$-subspace by at least $\arccos(1/\sqrt n)$. The construction is motivated by the hypothesis of Goreinov, Tyrtyshnikov and Zamarashkin that this value is the largest possible one for all $n > k > 0$. The subspaces are scaled star spaces of $2$-connected series-parallel graphs with $k+1$ vertices and $n$ edges, equipped with a particular positive edge weighting, while the largest principal angles take two values -- $\arccos(1 / \sqrt{n})$ and $π/2$, depending on whether a $k$-edge subgraph corresponding to a coordinate $k$-subspace is a spanning tree or not. For a fixed series-parallel graph, we also prove that the constructed weighting is the unique positive one, up to scaling, for which the corresponding $k$-subspace deviates from all coordinate $k$-subspaces by at least $\arccos(1 / \sqrt{n})$. |
| title | About subspaces the most deviating from the coordinate ones |
| topic | Numerical Analysis Combinatorics |
| url | https://arxiv.org/abs/2511.02387 |