Spectral Properties of Kernel Matrices in the Flat Limit∗
Total Page:16
File Type:pdf, Size:1020Kb

Load more
Recommended publications
-
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. -
Solutions to Math 53 Practice Second Midterm
Solutions to Math 53 Practice Second Midterm 1. (20 points) dx dy (a) (10 points) Write down the general solution to the system of equations dt = x+y; dt = −13x−3y in terms of real-valued functions. (b) (5 points) The trajectories of this equation in the (x; y)-plane rotate around the origin. Is this dx rotation clockwise or counterclockwise? If we changed the first equation to dt = −x + y, would it change the direction of rotation? (c) (5 points) Suppose (x1(t); y1(t)) and (x2(t); y2(t)) are two solutions to this equation. Define the Wronskian determinant W (t) of these two solutions. If W (0) = 1, what is W (10)? Note: the material for this part will be covered in the May 8 and May 10 lectures. 1 1 (a) The corresponding matrix is A = , with characteristic polynomial (1 − λ)(−3 − −13 −3 λ) + 13 = 0, so that λ2 + 2λ + 10 = 0, so that λ = −1 ± 3i. The corresponding eigenvector v to λ1 = −1 + 3i must be a solution to 2 − 3i 1 −13 −2 − 3i 1 so we can take v = Now we just need to compute the real and imaginary parts of 3i − 2 e3it cos(3t) + i sin(3t) veλ1t = e−t = e−t : (3i − 2)e3it −2 cos(3t) − 3 sin(3t) + i(3 cos(3t) − 2 sin(3t)) The general solution is thus expressed as cos(3t) sin(3t) c e−t + c e−t : 1 −2 cos(3t) − 3 sin(3t) 2 (3 cos(3t) − 2 sin(3t)) dx (b) We can examine the direction field. -
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. -
Linear Independence, the Wronskian, and Variation of Parameters
LINEAR INDEPENDENCE, THE WRONSKIAN, AND VARIATION OF PARAMETERS JAMES KEESLING In this post we determine when a set of solutions of a linear differential equation are linearly independent. We first discuss the linear space of solutions for a homogeneous differential equation. 1. Homogeneous Linear Differential Equations We start with homogeneous linear nth-order ordinary differential equations with general coefficients. The form for the nth-order type of equation is the following. dnx dn−1x (1) a (t) + a (t) + ··· + a (t)x = 0 n dtn n−1 dtn−1 0 It is straightforward to solve such an equation if the functions ai(t) are all constants. However, for general functions as above, it may not be so easy. However, we do have a principle that is useful. Because the equation is linear and homogeneous, if we have a set of solutions fx1(t); : : : ; xn(t)g, then any linear combination of the solutions is also a solution. That is (2) x(t) = C1x1(t) + C2x2(t) + ··· + Cnxn(t) is also a solution for any choice of constants fC1;C2;:::;Cng. Now if the solutions fx1(t); : : : ; xn(t)g are linearly independent, then (2) is the general solution of the differential equation. We will explain why later. What does it mean for the functions, fx1(t); : : : ; xn(t)g, to be linearly independent? The simple straightforward answer is that (3) C1x1(t) + C2x2(t) + ··· + Cnxn(t) = 0 implies that C1 = 0, C2 = 0, ::: , and Cn = 0 where the Ci's are arbitrary constants. This is the definition, but it is not so easy to determine from it just when the condition holds to show that a given set of functions, fx1(t); x2(t); : : : ; xng, is linearly independent. -
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. -
An Introduction to the HESSIAN
An introduction to the HESSIAN 2nd Edition Photograph of Ludwig Otto Hesse (1811-1874), circa 1860 https://upload.wikimedia.org/wikipedia/commons/6/65/Ludwig_Otto_Hesse.jpg See page for author [Public domain], via Wikimedia Commons I. As the students of the first two years of mathematical analysis well know, the Wronskian, Jacobian and Hessian are the names of three determinants and matrixes, which were invented in the nineteenth century, to make life easy to the mathematicians and to provide nightmares to the students of mathematics. II. I don’t know how the Hessian came to the mind of Ludwig Otto Hesse (1811-1874), a quiet professor father of nine children, who was born in Koenigsberg (I think that Koenigsberg gave birth to a disproportionate 1 number of famous men). It is possible that he was studying the problem of finding maxima, minima and other anomalous points on a bi-dimensional surface. (An alternative hypothesis will be presented in section X. ) While pursuing such study, in one variable, one first looks for the points where the first derivative is zero (if it exists at all), and then examines the second derivative at each of those points, to find out its “quality”, whether it is a maximum, a minimum, or an inflection point. The variety of anomalies on a bi-dimensional surface is larger than for a one dimensional line, and one needs to employ more powerful mathematical instruments. Still, also in two dimensions one starts by looking for points where the two first partial derivatives, with respect to x and with respect to y respectively, exist and are both zero. -
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 .........................