On the eigenvalues of Euclidean distance matrices

Abstract

Abstract. In this paper, the notion of equitable partitions (EP) is used to study the eigenvalues of Euclidean distance matrices (EDMs). In particular, EP is used to obtain the characteristic polynomials of regular EDMs and non-spherical centrally symmetric EDMs. The paper also presents methods for constructing cospectral EDMs and EDMs with exactly three distinct eigenvalues.Mathematical subject classification: 51K05, 15A18, 05C50.

Fast nonrigid 3D retrieval using modal space transform

“…(Such matrix has a single positive eigenvalue, and the rest are negative. See [1] and references therein) Hence a scale invariant spectrum can be defined as…”

A note on “Methods for constructing distance matrices and the inverse eigenvalue problem”

“…Let λ 1 > 0 ≥ λ 2 ≥ · · · ≥ λ n , n i=1 λ i = 0, be eigenvalues of the matrix M n , defined by (2). The matrix M n is an Euclidean distance matrix iff w T n e, given in (11), is nonnegative.…”

“…First, let us construct a symmetric nonnegative matrix with zero diagonal and eigenvalues −1, −200, −3000, −40000, −500000 and 543201, , by using (2). Note that the matrix D is not EDM, since the corresponding w T e = −0.00049, but its leading principal submatrix of size 4 is EDM.…”

“…Euclidean distance matrices are symmetric, have a zero diagonal (thus the sum of their eigenvalues is zero) and precisely one positive eigenvalue (for a nonzero matrix). Besides [1,2] not much on eigenvalues of Euclidean distance matrices is known. In the paper [7], a conjecture was stated, that for arbitrary numbers…”

On Euclidean distance matrices of graphs

“…where s is a vector such that s T e = 1 (see [1]). Since F is positive semidefinite, it can be written as F = X T X with X = diag( √ σ i ) U T , where F = U ΣU T is the singular value decomposition of F and Σ = diag(σ i ).…”