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Eigenvalues of Euclidean Distance Matrices and Rs-Majorization on R2
Archive of SID 46th Annual Iranian Mathematics Conference 25-28 August 2015 Yazd University 2 Talk Eigenvalues of Euclidean distance matrices and rs-majorization on R pp.: 1{4 Eigenvalues of Euclidean Distance Matrices and rs-majorization on R2 Asma Ilkhanizadeh Manesh∗ Department of Pure Mathematics, Vali-e-Asr University of Rafsanjan Alemeh Sheikh Hoseini Department of Pure Mathematics, Shahid Bahonar University of Kerman Abstract Let D1 and D2 be two Euclidean distance matrices (EDMs) with correspond- ing positive semidefinite matrices B1 and B2 respectively. Suppose that λ(A) = ((λ(A)) )n is the vector of eigenvalues of a matrix A such that (λ(A)) ... i i=1 1 ≥ ≥ (λ(A))n. In this paper, the relation between the eigenvalues of EDMs and those of the 2 corresponding positive semidefinite matrices respect to rs, on R will be investigated. ≺ Keywords: Euclidean distance matrices, Rs-majorization. Mathematics Subject Classification [2010]: 34B15, 76A10 1 Introduction An n n nonnegative and symmetric matrix D = (d2 ) with zero diagonal elements is × ij called a predistance matrix. A predistance matrix D is called Euclidean or a Euclidean distance matrix (EDM) if there exist a positive integer r and a set of n points p1, . , pn r 2 2 { } such that p1, . , pn R and d = pi pj (i, j = 1, . , n), where . denotes the ∈ ij k − k k k usual Euclidean norm. The smallest value of r that satisfies the above condition is called the embedding dimension. As is well known, a predistance matrix D is Euclidean if and 1 1 t only if the matrix B = − P DP with P = I ee , where I is the n n identity matrix, 2 n − n n × and e is the vector of all ones, is positive semidefinite matrix. -
Polynomial Approximation Algorithms for Belief Matrix Maintenance in Identity Management
Polynomial Approximation Algorithms for Belief Matrix Maintenance in Identity Management Hamsa Balakrishnan, Inseok Hwang, Claire J. Tomlin Dept. of Aeronautics and Astronautics, Stanford University, CA 94305 hamsa,ishwang,[email protected] Abstract— Updating probabilistic belief matrices as new might be constrained to some prespecified (but not doubly- observations arrive, in the presence of noise, is a critical part stochastic) row and column sums. This paper addresses the of many algorithms for target tracking in sensor networks. problem of updating belief matrices by scaling in the face These updates have to be carried out while preserving sum constraints, arising for example, from probabilities. This paper of uncertainty in the system and the observations. addresses the problem of updating belief matrices to satisfy For example, consider the case of the belief matrix for a sum constraints using scaling algorithms. We show that the system with three objects (labelled 1, 2 and 3). Suppose convergence behavior of the Sinkhorn scaling process, used that, at some instant, we are unsure about their identities for scaling belief matrices, can vary dramatically depending (tagged X, Y and Z) completely, and our belief matrix is on whether the prior unscaled matrix is exactly scalable or only almost scalable. We give an efficient polynomial-time algo- a 3 × 3 matrix with every element equal to 1/3. Let us rithm based on the maximum-flow algorithm that determines suppose the we receive additional information that object whether a given matrix is exactly scalable, thus determining 3 is definitely Z. Then our prior, but constraint violating the convergence properties of the Sinkhorn scaling process. -
Arxiv:1306.4805V3 [Math.OC] 6 Feb 2015 Used Greedy Techniques to Reorder Matrices
CONVEX RELAXATIONS FOR PERMUTATION PROBLEMS FAJWEL FOGEL, RODOLPHE JENATTON, FRANCIS BACH, AND ALEXANDRE D’ASPREMONT ABSTRACT. Seriation seeks to reconstruct a linear order between variables using unsorted, pairwise similarity information. It has direct applications in archeology and shotgun gene sequencing for example. We write seri- ation as an optimization problem by proving the equivalence between the seriation and combinatorial 2-SUM problems on similarity matrices (2-SUM is a quadratic minimization problem over permutations). The seriation problem can be solved exactly by a spectral algorithm in the noiseless case and we derive several convex relax- ations for 2-SUM to improve the robustness of seriation solutions in noisy settings. These convex relaxations also allow us to impose structural constraints on the solution, hence solve semi-supervised seriation problems. We derive new approximation bounds for some of these relaxations and present numerical experiments on archeological data, Markov chains and DNA assembly from shotgun gene sequencing data. 1. INTRODUCTION We study optimization problems written over the set of permutations. While the relaxation techniques discussed in what follows are applicable to a much more general setting, most of the paper is centered on the seriation problem: we are given a similarity matrix between a set of n variables and assume that the variables can be ordered along a chain, where the similarity between variables decreases with their distance within this chain. The seriation problem seeks to reconstruct this linear ordering based on unsorted, possibly noisy, pairwise similarity information. This problem has its roots in archeology [Robinson, 1951] and also has direct applications in e.g. -
Doubly Stochastic Matrices Whose Powers Eventually Stop
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Linear Algebra and its Applications 330 (2001) 25–30 www.elsevier.com/locate/laa Doubly stochastic matrices whose powers eventually stopୋ Suk-Geun Hwang a,∗, Sung-Soo Pyo b aDepartment of Mathematics Education, Kyungpook National University, Taegu 702-701, South Korea bCombinatorial and Computational Mathematics Center, Pohang University of Science and Technology, Pohang, South Korea Received 22 June 2000; accepted 14 November 2000 Submitted by R.A. Brualdi Abstract In this note we characterize doubly stochastic matrices A whose powers A, A2,A3,... + eventually stop, i.e., Ap = Ap 1 =···for some positive integer p. The characterization en- ables us to determine the set of all such matrices. © 2001 Elsevier Science Inc. All rights reserved. AMS classification: 15A51 Keywords: Doubly stochastic matrix; J-potent 1. Introduction Let R denote the real field. For positive integers m, n,letRm×n denote the set of all m × n matrices with real entries. As usual let Rn denote the set Rn×1.We call the members of Rn the n-vectors. The n-vector of 1’s is denoted by e,andthe identity matrix of order n is denoted by In. For two matrices A, B of the same size, let A B denote that all the entries of A − B are nonnegative. A matrix A is called nonnegative if A O. A nonnegative square matrix is called a doubly stochastic matrix if all of its row sums and column sums equal 1. -
On the Eigenvalues of Euclidean Distance Matrices
“main” — 2008/10/13 — 23:12 — page 237 — #1 Volume 27, N. 3, pp. 237–250, 2008 Copyright © 2008 SBMAC ISSN 0101-8205 www.scielo.br/cam On the eigenvalues of Euclidean distance matrices A.Y. ALFAKIH∗ Department of Mathematics and Statistics University of Windsor, Windsor, Ontario N9B 3P4, Canada E-mail: [email protected] 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 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. Mathematical subject classification: 51K05, 15A18, 05C50. Key words: Euclidean distance matrices, eigenvalues, equitable partitions, characteristic poly- nomial. 1 Introduction ( ) An n ×n nonzero matrix D = di j is called a Euclidean distance matrix (EDM) 1, 2,..., n r if there exist points p p p in some Euclidean space < such that i j 2 , ,..., , di j = ||p − p || for all i j = 1 n where || || denotes the Euclidean norm. i , ,..., Let p , i ∈ N = {1 2 n}, be the set of points that generate an EDM π π ( , ,..., ) D. An m-partition of D is an ordered sequence = N1 N2 Nm of ,..., nonempty disjoint subsets of N whose union is N. The subsets N1 Nm are called the cells of the partition. The n-partition of D where each cell consists #760/08. Received: 07/IV/08. Accepted: 17/VI/08. ∗Research supported by the Natural Sciences and Engineering Research Council of Canada and MITACS. -
Smith Normal Formal of Distance Matrix of Block Graphs∗†
Ann. of Appl. Math. 32:1(2016); 20-29 SMITH NORMAL FORMAL OF DISTANCE MATRIX OF BLOCK GRAPHS∗y Jing Chen1;2,z Yaoping Hou2 (1. The Center of Discrete Math., Fuzhou University, Fujian 350003, PR China; 2. School of Math., Hunan First Normal University, Hunan 410205, PR China) Abstract A connected graph, whose blocks are all cliques (of possibly varying sizes), is called a block graph. Let D(G) be its distance matrix. In this note, we prove that the Smith normal form of D(G) is independent of the interconnection way of blocks and give an explicit expression for the Smith normal form in the case that all cliques have the same size, which generalize the results on determinants. Keywords block graph; distance matrix; Smith normal form 2000 Mathematics Subject Classification 05C50 1 Introduction Let G be a connected graph (or strong connected digraph) with vertex set f1; 2; ··· ; ng. The distance matrix D(G) is an n × n matrix in which di;j = d(i; j) denotes the distance from vertex i to vertex j: Like the adjacency matrix and Lapla- cian matrix of a graph, D(G) is also an integer matrix and there are many results on distance matrices and their applications. For distance matrices, Graham and Pollack [10] proved a remarkable result that gives a formula of the determinant of the distance matrix of a tree depend- ing only on the number n of vertices of the tree. The determinant is given by det D = (−1)n−1(n − 1)2n−2: This result has attracted much interest in algebraic graph theory. -
Arxiv:1803.06211V1 [Math.NA] 16 Mar 2018
A NUMERICAL MODEL FOR THE CONSTRUCTION OF FINITE BLASCHKE PRODUCTS WITH PREASSIGNED DISTINCT CRITICAL POINTS CHRISTER GLADER AND RAY PORN¨ Abstract. We present a numerical model for determining a finite Blaschke product of degree n + 1 having n preassigned distinct critical points z1; : : : ; zn in the complex (open) unit disk D. The Blaschke product is uniquely determined up to postcomposition with conformal automor- phisms of D. The proposed method is based on the construction of a sparse nonlinear system where the data dependency is isolated to two vectors and on a certain transformation of the critical points. The effi- ciency and accuracy of the method is illustrated in several examples. 1. Introduction A finite Blaschke product of degree n is a rational function of the form n Y z − αj (1.1) B(z) = c ; c; α 2 ; jcj = 1; jα j < 1 ; 1 − α z j C j j=1 j which thereby has all its zeros in the open unit disc D, all poles outside the closed unit disc D and constant modulus jB(z)j = 1 on the unit circle T. The overbar in (1.1) and in the sequel stands for complex conjugation. The finite Blaschke products of degree n form a subset of the rational functions of degree n which are unimodular on T. These functions are given by all fractions n ~ a0 + a1 z + ::: + an z (1.2) B(z) = n ; a0; :::; an 2 C : an + an−1 z + ::: + a0 z An irreducible rational function of form (1.2) is a finite Blaschke product when all its zeros are in D. -
Distance Matrix and Laplacian of a Tree with Attached Graphs R.B
Linear Algebra and its Applications 411 (2005) 295–308 www.elsevier.com/locate/laa Distance matrix and Laplacian of a tree with attached graphs R.B. Bapat ∗ Indian Statistical Institute, Delhi Centre, 7 S.J.S.S. Marg, New Delhi 110 016, India Received 5 November 2003; accepted 22 June 2004 Available online 7 August 2004 Submitted by S. Hedayat Abstract A tree with attached graphs is a tree, together with graphs defined on its partite sets. We introduce the notion of incidence matrix, Laplacian and distance matrix for a tree with attached graphs. Formulas are obtained for the minors of the incidence matrix and the Laplacian, and for the inverse and the determinant of the distance matrix. The case when the attached graphs themselves are trees is studied more closely. Several known results, including the Matrix Tree theorem, are special cases when the tree is a star. The case when the attached graphs are paths is also of interest since it is related to the transportation problem. © 2004 Elsevier Inc. All rights reserved. AMS classification: 15A09; 15A15 Keywords: Tree; Distance matrix; Resistance distance; Laplacian matrix; Determinant; Transportation problem 1. Introduction Minors of matrices associated with a graph has been an area of considerable inter- est, starting with the celebrated Matrix Tree theorem of Kirchhoff which asserts that any cofactor of the Laplacian matrix equals the number of spanning trees in the ∗ Fax: +91 11 26856779. E-mail address: [email protected] 0024-3795/$ - see front matter 2004 Elsevier Inc. All rights reserved. doi:10.1016/j.laa.2004.06.017 296 R.B. -
Hypercubes Are Determined by Their Distance Spectra 3
HYPERCUBES ARE DETERMINED BY THEIR DISTANCE SPECTRA JACK H. KOOLEN♦, SAKANDER HAYAT♠, AND QUAID IQBAL♣ Abstract. We show that the d-cube is determined by the spectrum of its distance matrix. 1. Introduction For undefined terms, see next section. For integers n 2 and d 2, the Ham- ming graph H(d, n) has as vertex set, the d-tuples with elements≥ from≥0, 1, 2,...,n 1 , with two d-tuples are adjacent if and only if they differ only in one{ coordinate.− For} a positive integer d, the d-cube is the graph H(d, 2). In this paper, we show that the d-cube is determined by its distance spectrum, see Theorem 5.4. Observe that the d-cube has exactly three distinct distance eigenvalues. Note that the d-cube is not determined by its adjacency spectrum as the Hoffman graph [11] has the same adjacency spectrum as the 4-cube and hence the cartesian product of (d 4)-cube and the Hoffman graph has the same adjacency spectrum − as the d-cube (for d 4, where the 0-cube is just K1). This also implies that the complement of the d-cube≥ is not determined by its distance spectrum, if d 4. ≥ There are quite a few papers on matrices of graphs with a few distinct eigenvalues. An important case is, when the Seidel matrix of a graph has exactly two distinct eigenvalues is very much studied, see, for example [18]. Van Dam and Haemers [20] characterized the graphs whose Laplacian matrix has two distinct nonzero eigenval- ues. -
Distance Matrix of a Class of Completely Positive Graphs
Spec. Matrices 2020; 8:160–171 Research Article Open Access Joyentanuj Das, Sachindranath Jayaraman, and Sumit Mohanty* Distance Matrix of a Class of Completely Positive Graphs: Determinant and Inverse https://doi.org/10.1515/spma-2020-0109 Received February 6, 2020; accepted May 14, 2020 Abstract: A real symmetric matrix A is said to be completely positive if it can be written as BBt for some (not necessarily square) nonnegative matrix B. A simple graph G is called a completely positive graph if every matrix realization of G that is both nonnegative and positive semidenite is a completely positive matrix. Our aim in this manuscript is to compute the determinant and inverse (when it exists) of the distance matrix of a class of completely positive graphs. We compute a matrix R such that the inverse of the distance matrix of a class of completely positive graphs is expressed a linear combination of the Laplacian matrix, a rank one matrix of all ones and R. This expression is similar to the existing result for trees. We also bring out interesting spectral properties of some of the principal submatrices of R. Keywords: Completely positive graphs, Schur complement, Laplacian matrix, Distance matrix MSC: 05C12, 05C50 1 Introduction and Preliminaries Let G = (V, E) be a nite, connected, simple and undirected graph with V as the set of vertices and E ⊂ V × V as the set of edges. We write i ∼ j to indicate that the vertices i, j are adjacent in G. The degree of the vertex i is denoted by δi. A graph with n vertices is called complete, if each vertex of the graph is adjacent to every other vertex and is denoted by Kn. -
"Distance Measures for Graph Theory"
Distance measures for graph theory : Comparisons and analyzes of different methods Dissertation presented by Maxime DUYCK for obtaining the Master’s degree in Mathematical Engineering Supervisor(s) Marco SAERENS Reader(s) Guillaume GUEX, Bertrand LEBICHOT Academic year 2016-2017 Acknowledgments First, I would like to thank my supervisor Pr. Marco Saerens for his presence, his advice and his precious help throughout the realization of this thesis. Second, I would also like to thank Bertrand Lebichot and Guillaume Guex for agreeing to read this work. Next, I would like to thank my parents, all my family and my friends to have accompanied and encouraged me during all my studies. Finally, I would thank Malian De Ron for creating this template [65] and making it available to me. This helped me a lot during “le jour et la nuit”. Contents 1. Introduction 1 1.1. Context presentation .................................. 1 1.2. Contents .......................................... 2 2. Theoretical part 3 2.1. Preliminaries ....................................... 4 2.1.1. Networks and graphs .............................. 4 2.1.2. Useful matrices and tools ........................... 4 2.2. Distances and kernels on a graph ........................... 7 2.2.1. Notion of (dis)similarity measures ...................... 7 2.2.2. Kernel on a graph ................................ 8 2.2.3. The shortest-path distance .......................... 9 2.3. Kernels from distances ................................. 9 2.3.1. Multidimensional scaling ............................ 9 2.3.2. Gaussian mapping ............................... 9 2.4. Similarity measures between nodes .......................... 9 2.4.1. Katz index and its Leicht’s extension .................... 10 2.4.2. Commute-time distance and Euclidean commute-time distance .... 10 2.4.3. SimRank similarity measure ......................... -
Notes on Birkhoff-Von Neumann Decomposition of Doubly Stochastic Matrices Fanny Dufossé, Bora Uçar
Notes on Birkhoff-von Neumann decomposition of doubly stochastic matrices Fanny Dufossé, Bora Uçar To cite this version: Fanny Dufossé, Bora Uçar. Notes on Birkhoff-von Neumann decomposition of doubly stochastic matri- ces. Linear Algebra and its Applications, Elsevier, 2016, 497, pp.108–115. 10.1016/j.laa.2016.02.023. hal-01270331v6 HAL Id: hal-01270331 https://hal.inria.fr/hal-01270331v6 Submitted on 23 Apr 2016 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Notes on Birkhoff-von Neumann decomposition of doubly stochastic matrices Fanny Dufoss´ea, Bora U¸carb,∗ aInria Lille, Nord Europe, 59650, Villeneuve d'Ascq, France bLIP, UMR5668 (CNRS - ENS Lyon - UCBL - Universit´ede Lyon - INRIA), Lyon, France Abstract Birkhoff-von Neumann (BvN) decomposition of doubly stochastic matrices ex- presses a double stochastic matrix as a convex combination of a number of permutation matrices. There are known upper and lower bounds for the num- ber of permutation matrices that take part in the BvN decomposition of a given doubly stochastic matrix. We investigate the problem of computing a decom- position with the minimum number of permutation matrices and show that the associated decision problem is strongly NP-complete.