About subspaces the most deviating from the coordinate ones

Abstract

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 $\pi/2$, 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 $\arccos(1 / \sqrt{n})$.