Euclidean Distance Matrix
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Parametrizations of K-Nonnegative Matrices
Parametrizations of k-Nonnegative Matrices Anna Brosowsky, Neeraja Kulkarni, Alex Mason, Joe Suk, Ewin Tang∗ October 2, 2017 Abstract Totally nonnegative (positive) matrices are matrices whose minors are all nonnegative (positive). We generalize the notion of total nonnegativity, as follows. A k-nonnegative (resp. k-positive) matrix has all minors of size k or less nonnegative (resp. positive). We give a generating set for the semigroup of k-nonnegative matrices, as well as relations for certain special cases, i.e. the k = n − 1 and k = n − 2 unitriangular cases. In the above two cases, we find that the set of k-nonnegative matrices can be partitioned into cells, analogous to the Bruhat cells of totally nonnegative matrices, based on their factorizations into generators. We will show that these cells, like the Bruhat cells, are homeomorphic to open balls, and we prove some results about the topological structure of the closure of these cells, and in fact, in the latter case, the cells form a Bruhat-like CW complex. We also give a family of minimal k-positivity tests which form sub-cluster algebras of the total positivity test cluster algebra. We describe ways to jump between these tests, and give an alternate description of some tests as double wiring diagrams. 1 Introduction A totally nonnegative (respectively totally positive) matrix is a matrix whose minors are all nonnegative (respectively positive). Total positivity and nonnegativity are well-studied phenomena and arise in areas such as planar networks, combinatorics, dynamics, statistics and probability. The study of total positivity and total nonnegativity admit many varied applications, some of which are explored in “Totally Nonnegative Matrices” by Fallat and Johnson [5]. -
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. -
Realizing Euclidean Distance Matrices by Sphere Intersection
Realizing Euclidean distance matrices by sphere intersection Jorge Alencara,∗, Carlile Lavorb, Leo Libertic aInstituto Federal de Educa¸c~ao,Ci^enciae Tecnologia do Sul de Minas Gerais, Inconfidentes, MG, Brazil bUniversity of Campinas (IMECC-UNICAMP), 13081-970, Campinas-SP, Brazil cCNRS LIX, Ecole´ Polytechnique, 91128 Palaiseau, France Abstract This paper presents the theoretical properties of an algorithm to find a re- alization of a (full) n × n Euclidean distance matrix in the smallest possible embedding dimension. Our algorithm performs linearly in n, and quadratically in the minimum embedding dimension, which is an improvement w.r.t. other algorithms. Keywords: Distance geometry, sphere intersection, euclidean distance matrix, embedding dimension. 2010 MSC: 00-01, 99-00 1. Introduction Euclidean distance matrices and sphere intersection have a strong mathe- matical importance [1, 2, 3, 4, 5, 6], in addition to many applications, such as navigation problems, molecular and nanostructure conformation, network 5 localization, robotics, as well as other problems of distance geometry [7, 8, 9]. Before defining precisely the problem of interest, we need the formal defini- tion for Euclidean distance matrices. Let D be a n × n symmetric hollow (i.e., with zero diagonal) matrix with non-negative elements. We say that D is a (squared) Euclidean Distance Matrix (EDM) if there are points x1; x2; : : : ; xn 2 RK (for a positive integer K) such that 2 D(i; j) = Dij = kxi − xjk ; i; j 2 f1; : : : ; ng; where k · k denotes the Euclidean norm. The smallest K for which such a set of points exists is called the embedding dimension of D, denoted by dim(D). -
1 Euclidean Vector Space and Euclidean Affi Ne Space
Profesora: Eugenia Rosado. E.T.S. Arquitectura. Euclidean Geometry1 1 Euclidean vector space and euclidean a¢ ne space 1.1 Scalar product. Euclidean vector space. Let V be a real vector space. De…nition. A scalar product is a map (denoted by a dot ) V V R ! (~u;~v) ~u ~v 7! satisfying the following axioms: 1. commutativity ~u ~v = ~v ~u 2. distributive ~u (~v + ~w) = ~u ~v + ~u ~w 3. ( ~u) ~v = (~u ~v) 4. ~u ~u 0, for every ~u V 2 5. ~u ~u = 0 if and only if ~u = 0 De…nition. Let V be a real vector space and let be a scalar product. The pair (V; ) is said to be an euclidean vector space. Example. The map de…ned as follows V V R ! (~u;~v) ~u ~v = x1x2 + y1y2 + z1z2 7! where ~u = (x1; y1; z1), ~v = (x2; y2; z2) is a scalar product as it satis…es the …ve properties of a scalar product. This scalar product is called standard (or canonical) scalar product. The pair (V; ) where is the standard scalar product is called the standard euclidean space. 1.1.1 Norm associated to a scalar product. Let (V; ) be a real euclidean vector space. De…nition. A norm associated to the scalar product is a map de…ned as follows V kk R ! ~u ~u = p~u ~u: 7! k k Profesora: Eugenia Rosado, E.T.S. Arquitectura. Euclidean Geometry.2 1.1.2 Unitary and orthogonal vectors. Orthonormal basis. Let (V; ) be a real euclidean vector space. De…nition. -
Tight Frames and Their Symmetries
Technical Report 9 December 2003 Tight Frames and their Symmetries Richard Vale, Shayne Waldron Department of Mathematics, University of Auckland, Private Bag 92019, Auckland, New Zealand e–mail: [email protected] (http:www.math.auckland.ac.nz/˜waldron) e–mail: [email protected] ABSTRACT The aim of this paper is to investigate symmetry properties of tight frames, with a view to constructing tight frames of orthogonal polynomials in several variables which share the symmetries of the weight function, and other similar applications. This is achieved by using representation theory to give methods for constructing tight frames as orbits of groups of unitary transformations acting on a given finite-dimensional Hilbert space. Along the way, we show that a tight frame is determined by its Gram matrix and discuss how the symmetries of a tight frame are related to its Gram matrix. We also give a complete classification of those tight frames which arise as orbits of an abelian group of symmetries. Key Words: Tight frames, isometric tight frames, Gram matrix, multivariate orthogonal polynomials, symmetry groups, harmonic frames, representation theory, wavelets AMS (MOS) Subject Classifications: primary 05B20, 33C50, 20C15, 42C15, sec- ondary 52B15, 42C40 0 1. Introduction u1 u2 u3 2 The three equally spaced unit vectors u1, u2, u3 in IR provide the following redundant representation 2 3 f = f, u u , f IR2, (1.1) 3 h ji j ∀ ∈ j=1 X which is the simplest example of a tight frame. Such representations arose in the study of nonharmonic Fourier series in L2(IR) (see Duffin and Schaeffer [DS52]) and have recently been used extensively in the theory of wavelets (see, e.g., Daubechies [D92]). -
Chapter Four Determinants
Chapter Four Determinants In the first chapter of this book we considered linear systems and we picked out the special case of systems with the same number of equations as unknowns, those of the form T~x = ~b where T is a square matrix. We noted a distinction between two classes of T ’s. While such systems may have a unique solution or no solutions or infinitely many solutions, if a particular T is associated with a unique solution in any system, such as the homogeneous system ~b = ~0, then T is associated with a unique solution for every ~b. We call such a matrix of coefficients ‘nonsingular’. The other kind of T , where every linear system for which it is the matrix of coefficients has either no solution or infinitely many solutions, we call ‘singular’. Through the second and third chapters the value of this distinction has been a theme. For instance, we now know that nonsingularity of an n£n matrix T is equivalent to each of these: ² a system T~x = ~b has a solution, and that solution is unique; ² Gauss-Jordan reduction of T yields an identity matrix; ² the rows of T form a linearly independent set; ² the columns of T form a basis for Rn; ² any map that T represents is an isomorphism; ² an inverse matrix T ¡1 exists. So when we look at a particular square matrix, the question of whether it is nonsingular is one of the first things that we ask. This chapter develops a formula to determine this. (Since we will restrict the discussion to square matrices, in this chapter we will usually simply say ‘matrix’ in place of ‘square matrix’.) More precisely, we will develop infinitely many formulas, one for 1£1 ma- trices, one for 2£2 matrices, etc. -
Week 8-9. Inner Product Spaces. (Revised Version) Section 3.1 Dot Product As an Inner Product
Math 2051 W2008 Margo Kondratieva Week 8-9. Inner product spaces. (revised version) Section 3.1 Dot product as an inner product. Consider a linear (vector) space V . (Let us restrict ourselves to only real spaces that is we will not deal with complex numbers and vectors.) De¯nition 1. An inner product on V is a function which assigns a real number, denoted by < ~u;~v> to every pair of vectors ~u;~v 2 V such that (1) < ~u;~v>=< ~v; ~u> for all ~u;~v 2 V ; (2) < ~u + ~v; ~w>=< ~u;~w> + < ~v; ~w> for all ~u;~v; ~w 2 V ; (3) < k~u;~v>= k < ~u;~v> for any k 2 R and ~u;~v 2 V . (4) < ~v;~v>¸ 0 for all ~v 2 V , and < ~v;~v>= 0 only for ~v = ~0. De¯nition 2. Inner product space is a vector space equipped with an inner product. Pn It is straightforward to check that the dot product introduces by ~u ¢ ~v = j=1 ujvj is an inner product. You are advised to verify all the properties listed in the de¯nition, as an exercise. The dot product is also called Euclidian inner product. De¯nition 3. Euclidian vector space is Rn equipped with Euclidian inner product < ~u;~v>= ~u¢~v. De¯nition 4. A square matrix A is called positive de¯nite if ~vT A~v> 0 for any vector ~v 6= ~0. · ¸ 2 0 Problem 1. Show that is positive de¯nite. 0 3 Solution: Take ~v = (x; y)T . Then ~vT A~v = 2x2 + 3y2 > 0 for (x; y) 6= (0; 0). -
On Asymmetric Distances
Analysis and Geometry in Metric Spaces Research Article • DOI: 10.2478/agms-2013-0004 • AGMS • 2013 • 200-231 On Asymmetric Distances Abstract ∗ In this paper we discuss asymmetric length structures and Andrea C. G. Mennucci asymmetric metric spaces. A length structure induces a (semi)distance function; by using Scuola Normale Superiore, Piazza dei Cavalieri 7, 56126 Pisa, Italy the total variation formula, a (semi)distance function induces a length. In the first part we identify a topology in the set of paths that best describes when the above operations are idempo- tent. As a typical application, we consider the length of paths Received 24 March 2013 defined by a Finslerian functional in Calculus of Variations. Accepted 15 May 2013 In the second part we generalize the setting of General metric spaces of Busemann, and discuss the newly found aspects of the theory: we identify three interesting classes of paths, and compare them; we note that a geodesic segment (as defined by Busemann) is not necessarily continuous in our setting; hence we present three different notions of intrinsic metric space. Keywords Asymmetric metric • general metric • quasi metric • ostensible metric • Finsler metric • run–continuity • intrinsic metric • path metric • length structure MSC: 54C99, 54E25, 26A45 © Versita sp. z o.o. 1. Introduction Besides, one insists that the distance function be symmetric, that is, d(x; y) = d(y; x) (This unpleasantly limits many applications [...]) M. Gromov ([12], Intr.) The main purpose of this paper is to study an asymmetric metric theory; this theory naturally generalizes the metric part of Finsler Geometry, much as symmetric metric theory generalizes the metric part of Riemannian Geometry. -
Glossary of Linear Algebra Terms
INNER PRODUCT SPACES AND THE GRAM-SCHMIDT PROCESS A. HAVENS 1. The Dot Product and Orthogonality 1.1. Review of the Dot Product. We first recall the notion of the dot product, which gives us a familiar example of an inner product structure on the real vector spaces Rn. This product is connected to the Euclidean geometry of Rn, via lengths and angles measured in Rn. Later, we will introduce inner product spaces in general, and use their structure to define general notions of length and angle on other vector spaces. Definition 1.1. The dot product of real n-vectors in the Euclidean vector space Rn is the scalar product · : Rn × Rn ! R given by the rule n n ! n X X X (u; v) = uiei; viei 7! uivi : i=1 i=1 i n Here BS := (e1;:::; en) is the standard basis of R . With respect to our conventions on basis and matrix multiplication, we may also express the dot product as the matrix-vector product 2 3 v1 6 7 t î ó 6 . 7 u v = u1 : : : un 6 . 7 : 4 5 vn It is a good exercise to verify the following proposition. Proposition 1.1. Let u; v; w 2 Rn be any real n-vectors, and s; t 2 R be any scalars. The Euclidean dot product (u; v) 7! u · v satisfies the following properties. (i:) The dot product is symmetric: u · v = v · u. (ii:) The dot product is bilinear: • (su) · v = s(u · v) = u · (sv), • (u + v) · w = u · w + v · w. -
Arxiv:1701.03378V1 [Math.RA] 12 Jan 2017 Setao Apihe Rusadfe Probability” Free and Groups T Lamplighter by on Supported Technology
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by TUGraz OPEN Library Linearizing the Word Problem in (some) Free Fields Konrad Schrempf∗ January 13, 2017 Abstract We describe a solution of the word problem in free fields (coming from non- commutative polynomials over a commutative field) using elementary linear algebra, provided that the elements are given by minimal linear representa- tions. It relies on the normal form of Cohn and Reutenauer and can be used more generally to (positively) test rational identities. Moreover we provide a construction of minimal linear representations for the inverse of non-zero elements. Keywords: word problem, minimal linear representation, linearization, realiza- tion, admissible linear system, rational series AMS Classification: 16K40, 16S10, 03B25, 15A22 Introduction arXiv:1701.03378v1 [math.RA] 12 Jan 2017 Free (skew) fields arise as universal objects when it comes to embed the ring of non-commutative polynomials, that is, polynomials in (a finite number of) non- commuting variables, into a skew field [Coh85, Chapter 7]. The notion of “free fields” goes back to Amitsur [Ami66]. A brief introduction can be found in [Coh03, Section 9.3], for details we refer to [Coh95, Section 6.4]. In the present paper we restrict the setting to commutative ground fields, as a special case. See also [Rob84]. In [CR94], Cohn and Reutenauer introduced a normal form for elements in free fields in order to extend results from the theory of formal languages. In particular they characterize minimality of linear representations in terms of linear independence of ∗Contact: [email protected], Department of Discrete Mathematics (Noncommutative Structures), Graz University of Technology. -
QUADRATIC FORMS and DEFINITE MATRICES 1.1. Definition of A
QUADRATIC FORMS AND DEFINITE MATRICES 1. DEFINITION AND CLASSIFICATION OF QUADRATIC FORMS 1.1. Definition of a quadratic form. Let A denote an n x n symmetric matrix with real entries and let x denote an n x 1 column vector. Then Q = x’Ax is said to be a quadratic form. Note that a11 ··· a1n . x1 Q = x´Ax =(x1...xn) . xn an1 ··· ann P a1ixi . =(x1,x2, ··· ,xn) . P anixi 2 (1) = a11x1 + a12x1x2 + ... + a1nx1xn 2 + a21x2x1 + a22x2 + ... + a2nx2xn + ... + ... + ... 2 + an1xnx1 + an2xnx2 + ... + annxn = Pi ≤ j aij xi xj For example, consider the matrix 12 A = 21 and the vector x. Q is given by 0 12x1 Q = x Ax =[x1 x2] 21 x2 x1 =[x1 +2x2 2 x1 + x2 ] x2 2 2 = x1 +2x1 x2 +2x1 x2 + x2 2 2 = x1 +4x1 x2 + x2 1.2. Classification of the quadratic form Q = x0Ax: A quadratic form is said to be: a: negative definite: Q<0 when x =06 b: negative semidefinite: Q ≤ 0 for all x and Q =0for some x =06 c: positive definite: Q>0 when x =06 d: positive semidefinite: Q ≥ 0 for all x and Q = 0 for some x =06 e: indefinite: Q>0 for some x and Q<0 for some other x Date: September 14, 2004. 1 2 QUADRATIC FORMS AND DEFINITE MATRICES Consider as an example the 3x3 diagonal matrix D below and a general 3 element vector x. 100 D = 020 004 The general quadratic form is given by 100 x1 0 Q = x Ax =[x1 x2 x3] 020 x2 004 x3 x1 =[x 2 x 4 x ] x2 1 2 3 x3 2 2 2 = x1 +2x2 +4x3 Note that for any real vector x =06 , that Q will be positive, because the square of any number is positive, the coefficients of the squared terms are positive and the sum of positive numbers is always positive.