The first eigenvector of a distance matrix is nearly constant

DOI10.1016/J.DISC.2022.113291arXiv2205.15920MaRDI QIDQ6400712

Publication date: 31 May 2022

Abstract: Let

x_{1}, d o t s, x_{n}

be points in a metric space and define the distance matrix

D i n m a t h b b R^{n i m e s n}

by

D_{i j} = d (x_{i}, x_{j})

. The Perron-Frobenius Theorem implies that there is an eigenvector

v i n m a t h b b 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

m a t h b b 1 i n m a t h b b R^{n}

is large

leftlangle v, mathbb{1} ight angle geq frac{1}{sqrt{2}} cdot | v|_{ell^2} cdot |mathbb{1} |_{ell^2}

and that each entry satisfies

v_{i} g e q | v |_{e l l^{2}} / s q r t 4 n

. Both inequalities are sharp.

Mathematics Subject Classification ID

Geometry and structure of normed linear spaces (46B20) Graphs and linear algebra (matrices, eigenvalues, etc.) (05C50) Eigenvalues, singular values, and eigenvectors (15A18)

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