The first eigenvector of a distance matrix is nearly constant

TL;DR

This paper proves that the dominant eigenvector of a distance matrix in a metric space is nearly constant, with specific bounds on its entries and inner product with the all-ones vector.

Contribution

It establishes sharp bounds showing the near-constancy of the principal eigenvector of a distance matrix, extending Perron-Frobenius insights to metric space contexts.

Findings

The eigenvector's inner product with the all-ones vector is large.

Each entry of the eigenvector is bounded below by a constant times its norm.

Both bounds are proven to be sharp.

Abstract

Let $x_1, \dots, x_n$ be points in a metric space and define the distance matrix $D \in \mathbb{R}^{n \times n}$ by ${D}_{ij} = d(x_i, x_j)$. The Perron-Frobenius Theorem implies that there is an eigenvector $v \in \mathbb{R}^n_{}$ with non-negative entries associated to the largest eigenvalue. We prove that this eigenvector is nearly constant in the sense that the inner product with the constant vector $\mathbb{1} \in \mathbb{R}^n$ is large $$ \left\langle v, \mathbb{1} \right\rangle \geq \frac{1}{\sqrt{2}} \cdot \| v\|_{\ell^2} \cdot \|\mathbb{1} \|_{\ell^2}$$ and that each entry satisfies $v_i \geq \|v\|_{\ell^2}/\sqrt{4n}$. Both inequalities are sharp.