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

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## Abstract

In this paper, a symmetric nonnegative matrix with zero diagonal and given spectrum, where exactly one of the eigenvalues is positive, is constructed. This solves the symmetric nonnegative eigenvalue problem (SNIEP) for such a spectrum. The construction is based on the idea from the paper Hayden, Reams, Wells, “Methods for constructing distance matrices and the inverse eigenvalue problem”. Some results of this paper are enhanced. The construction is applied for the solution of the inverse eigenvalue problem for Euclidean distance matrices, under some assumptions on the eigenvalues.

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... Euclidean distance matrices were introduced by Menger in 1928, later they were studied by Schoenberg [22], Gower [9], and other authors. In recent years many new results were obtained (see [13,15] and the references therein). EDMs have many interesting properties and are used in various applications in linear algebra, graph theory, bioinformatics, e.g., where frequently a question arises, what can be said about a configuration of points x i , if only distances between them are known. ...
... A nonzero EDM has exactly one positive eigenvalue and the sum of its eigenvalues is zero. It is conjectured that there always exists a solution of the inverse eigenvalue problem, i.e., to prove that any set of numbers that meet these conditions is a spectrum of an EDM (see [15,17], e.g). ...
Article
In this paper a relation between graph distance matrices of the star graph and its generalizations and Euclidean distance matrices is considered. It is proven that distance matrices of certain families of graphs are circum Euclidean. Their spectrum and generating points are given in a closed form.
... Distance matrices and its eigenvalues are studied in several papers such as [5,6,8]. ...
... In the paper [8], G. Jaklič and J. Modic offered a method for constructing a symmetric nonnegative matrix with zero diagonal and eigenvalues λ i , where n i=1 λ i = 0 and λ 1 > 0 > λ 2 · · · λ n , then they survey the inverse eigenvalue problem for Euclidean distance matrices, which are a subclass of such matrices. Although this method is useful for solving many problem of inverse eigenvalue problems of distance matrices, but it has some constraints, for example large matrix dimension, and there is no solution for some special spectrum. ...
... Theorem 4.7. [11] Let n ≥ ℓ ≥ 2 and let λ 1 ≤ · · · ≤ λ n−ℓ+1 < λ n−ℓ+2 = · · · = λ n−1 = 0 < λ n be real numbers such that n i=1 λ i = 0. Then there is a nonnegative matrix A ∈ S 0 (K ℓ−1 ∨ K n−ℓ+1 ) such that spec(A) = {λ 1 , . . . , λ n−ℓ+1 , 0 (ℓ−2) , λ n }. ...
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A hollow matrix described by a graph $G$ is a real symmetric matrix having all diagonal entries equal to zero and with the off-diagonal entries governed by the adjacencies in $G$. For a given graph $G$, the determination of all possible spectra of matrices associated with $G$ is the hollow inverse eigenvalue problem for $G$. Solutions to the hollow inverse eigenvalue problems for paths and complete bipartite graphs are presented. Results for related subproblems such as possible ordered multiplicity lists, maximum multiplicity of an eigenvalue, and minimum number of distinct eigenvalues are presented for additional families of graphs.
... For instance, T.L. Hayden, R. Reams and J. Wells have solved the inverse eigenvalue problem for Euclidean distance matrices of order n = 3, 4, 5, 6 , and any n for which there exists a Hadamard matrix and also they solved this problem: If for n ∈ ℕ there exists a Hadamard matrix of order n, then there is an (n + 1) × (n + 1) and an (n + 2) × (n + 2) distance matrix with eigenvalues which hold under special conditions for n ⩽ 16 [13]. In the paper [14], Jaklic̆ and Modic offered a method for constructing a symmetric nonnegative matrix with zero diagonal and eigenvalues i , where ∑ n i=1 i = 0 and 1 > 0 > 2 ⩾ ⋯ ⩾ n , then they survey the inverse eigenvalue problem for Euclidean distance matrices, which are a subclass of such matrices. Nazari and Mahdinasab solved this problem without using any Hadamard matrix [15]. ...
Article
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In this paper, at first, for a given set of real numbers with only one positive number, and in continue for a given set of real numbers in special conditions, we construct a symmetric nonnegative matrix such that the given set is its spectrum.
... They were studied by Schoenberg [13], Young and Householder [14], Gower [4], and many other authors. In recent years many new results were obtained (see [5,7,8,11] and the references therein). ...
Article
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In this paper, a relation between graph distance matrices and Euclidean distance matrices (EDM) is considered. Graphs, for which the distance matrix is not an EDM (NEDM-graphs), are studied. All simple connected non-isomorphic graphs on n <= 8 nodes are analysed and a characterization of the smallest NEDM-graphs, i.e., the minimal forbidden subgraphs, is given. It is proven that bipartite graphs and some subdivisions of the smallest NEDM-graphs are NEDM-graphs, too.
... A nonzero EDM has only one positive eigenvalue λ 1 , and the sum of its eigenvalues is zero. It is conjectured that any set of numbers that meet these conditions can be a spectrum of an EDM (see, e.g., [10,11]). ...
Article
In this paper, a relation between graph distance matrices and Euclidean distance matrices (EDM) is considered. It is proven that distance matrices of paths and cycles are EDMs. The proofs are constructive and the generating points of studied EDMs are given in a closed form. A generalization to weighted graphs (networks) is tackled.
Chapter
This chapter provides an introduction to Euclidean distance matrices (EDMs). Our primary focus is on various characterizations and basic properties of EDMs. The chapter also discusses methods to construct new EDMs from old ones, and presents some EDM necessary and sufficient inequalities. It also provides a discussion of the Cayley–Menger matrix and Schoenberg Transformations.
Chapter
The focus of this chapter is on the eigenvalues of EDMs. In the first part, we present a characterization of the column space of an EDM D. This characterization is then used to express the eigenvalues of D in terms of the eigenvalues of its Gram matrix $$B =\mathcal{ T}(D) = -JDJ/2$$. In case of regular and nonspherical centrally symmetric EDMs, the same result can also be obtained by using the notion of equitable partition. In the second part, we discuss some other topics related to eigenvalues such as: a method for constructing nonisomorphic cospectral EDMs; the connection between EDMs, graphs, and combinatorial designs; EDMs with exactly two or three distinct eigenvalues and the EDM inverse eigenvalue problem.
Article
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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 poly-nomials 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.
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The basic goal of an inverse eigenvalue problem is to reconstruct the physical parameters of a certain system from the knowledge or desire of its dynamical behavior. Depending on the application, inverse eigenvalue problems appear in many different forms. This book discusses the fundamental questions, some known results, many applications, mathematical properties, a variety of numerical techniques, as well as several open problems.
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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 nonspherical centrally symmetric EDMs. The paper also presents methods for constructing cospectral EDMs and EDMs with exactly three distinct eigenvalues.
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Given two vectors $a,\lambda \in R^n$, the Schur–Horn theorem states that a majorizes $\lambda$ if and only if there exists a Hermitian matrix H with eigenvalues $\lambda$ and diagonal entries a. While the theory is regarded as classical by now, the known proof is not constructive. To construct a Hermitian matrix from its diagonal entries and eigenvalues therefore becomes an interesting and challenging inverse eigenvalue problem. Two algorithms for determining the matrix numerically are proposed in this paper. The lift and projection method is an iterative method that involves an interesting application of the Wielandt–Hoffman theorem. The projected gradient method is a continuous method that, besides its easy implementation, offers a new proof of existence because of its global convergence property.
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Let σ=(λ1,…,λn) be the spectrum of a nonnegative symmetric matrix A with the Perron eigenvalue λ1, a diagonal entry c and let τ=(μ1,…,μm) be the spectrum of a nonnegative symmetric matrix B with the Perron eigenvalue μ1. We show how to construct a nonnegative symmetric matrix C with the spectrum(λ1+max{0,μ1-c},λ2,…,λn,μ2,…,μm).
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Let and be two distance matrices. We provide necessary conditions on in order thatbe a distance matrix. We then show that it is always possible to border an n×n distance matrix, with certain scalar multiples of its Perron eigenvector, to construct an (n+1)×(n+1) distance matrix. We also give necessary and sufficient conditions for two principal distance matrix blocks and be used to form a distance matrix as above, where Z is a scalar multiple of a rank one matrix, formed from their Perron eigenvectors. Finally, we solve the inverse eigenvalue problem for distance matrices in certain special cases, including n=3,4,5,6, any n for which there exists a Hadamard matrix, and some other cases.
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We present a characterization of the nullspace and the rangespace of a Euclidean distance matrix (EDM) D in terms of the vector of all ones, and in terms of the Gale subspace G(D) and the realization matrix P corresponding to D. This characterization is then used to compute the characteristic polynomial of D. We also present some results concerning EDMs generated by regular figures and EDMs generated by centrally symmetric points.
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In this paper properties of cell matrices are studied. A determinant of such a matrix is given in a closed form. In the proof a general method for determining a determinant of a symbolic matrix with polynomial entries, based on multivariate polynomial Lagrange interpolation, is outlined. It is shown that a cell matrix of size n>1 has exactly one positive eigenvalue. Using this result it is proven that cell matrices are (Circum-)Euclidean Distance Matrices ((C)EDM), and their generalization, k-cell matrices, are CEDM under certain natural restrictions. A characterization of k-cell matrices is outlined.