About subspaces the most deviating from the coordinate ones

TL;DR

The paper investigates the maximal deviation, under the largest principal angle, between a $k$-dimensional subspace of $\mathds{R}^n$ and all coordinate subspaces of the same dimension, addressing the GTZ1997 bound $\arccos(1/\sqrt{n})$ on this distance. It introduces graph-theoretic realizations of extremal subspaces via star-spaces of directed 2-connected series-parallel graphs with graph-induced edge weights, formalized through incidence matrices $B$ and weight matrices $W$, and analyzes these using the transfer current matrix $Y$ and Laplacian pseudoinverses. The main result proves that the column space of $W^{1/2} B^T$ has a distance at least $\arccos(1/\sqrt{n})$ from any coordinate subspace, with the bound tight and unattainable to improve; the proof hinges on eigenvalue analysis of tree subspaces and duality arguments. The work also bridges linear algebra with graph theory and probability, suggesting broader applicability of star-spaces and transfer-current concepts in related domains.

Abstract

Taking the largest principal angle as the distance function between same dimensional nontrivial linear subspaces in $\mathds{R}^n$, we describe the class of subspaces deviating from all the coordinate ones by at least $\arccos(1 / \sqrt{n})$. This study compliments and is motivated by the long-standing hypothesis put forward in \cite{GTZ1997} and essentially stating that so-defined distance to the closest coordinate subspace cannot exceed $\arccos(1 / \sqrt{n})$. In this context, the subspaces presented here claim to be the extremal ones. Realized as the star spaces of all nontrivial 2-connected series-parallel graphs with certain edge weights and arbitrary edge directions, the given subspaces may be of interest beyond numerical linear algebra within which the original problem was formulated.

Figures (4)

• Figure 1: A series-parallel graph and its decomposition tree for terminal vertices $a$ and $b$.
• Figure 2: A series-parallel graph and its graph-induced edge weights.
• Figure 3: A directed series-parallel graph, its spanning tree (in red) and the induced edge coefficients.
• Figure 4: Graph $G$ and the layout of $\tau$ (in red) for the case $d = 2$.

Theorems & Definitions (14)

• Definition 2.1
• Definition 2.2
• Definition 2.3
• Remark
• Remark
• Definition 2.4
• Definition 3.1
• Proposition 3.1
• proof
• Remark