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Spectral Properties of the Gramian of Finite Ultrametric Spaces (2504.14840v1)

Published 21 Apr 2025 in math.FA

Abstract: The concept of $p$-negative type is such that a metric space $(X,d_{X})$ has $p$-negative type if and only if $(X,d_{X}{p/2})$ embeds isometrically into a Hilbert space. If $X={x_{0},x_{1},\dots,x_{n}}$ then the $p$-negative type of $X$ is intimately related to the Gramian matrix $G_{p}=(g_{ij}){i,j=1}{n}$ where $g{ij}=\frac{1}{2}(d_{X}(x_{i},x_{0}){p}+d_{X}(x_{j},x_{0}){p}-d_{X}(x_{i},x_{j}){p})$. In particular, $X$ has strict $p$-negative type if and only if $G_{p}$ is strictly positive semidefinite. As such, a natural measure of the degree of strictness of $p$-negative type that $X$ possesses is the minimum eigenvalue of the Gramian $\lambda_{min}(G_{p})$. In this article we compute the minimum eigenvalue of the Gramian of a finite ultrametric space. Namely, if $X$ is a finite ultrametric space with minimum nonzero distance $\alpha_{1}$ then we show that $\lambda_{min}(G_{p})=\alpha_{1}{p}/2$. We also provide a description of the corresponding eigenspace.

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