The first eigenvector of a distance matrix is nearly constant

2022/05/31 by Steinerberger, Stefan · 1 citation
#Combinatorics (math.CO) #FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.2205.15920

Abstract

Let x₁, …, x_n be points in a metric space and define the distance matrix D ∈ ℝ^{n × n} by D_ij = d(x_i, x_j). The Perron-Frobenius Theorem implies that there is an eigenvector v ∈ ℝⁿ 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 \mathbb1 ∈ ℝⁿ is large ⟨ v, \mathbb1 ⟩ ≥ (1)/(√(2)) ⋅ ‖ v‖_ℓ² ⋅ ‖\mathbb1 ‖_ℓ² and that each entry satisfies v_i ≥ ‖v‖_ℓ²/√(4n). Both inequalities are sharp.

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• Potential Theory and the Boundary of Combinatorial Graphs

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