On Orthogonal Matrices with Zero Diagonal
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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). -
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. -
Codes and Modules Associated with Designs and T-Uniform Hypergraphs Richard M
Codes and modules associated with designs and t-uniform hypergraphs Richard M. Wilson California Institute of Technology (1) Smith and diagonal form (2) Solutions of linear equations in integers (3) Square incidence matrices (4) A chain of codes (5) Self-dual codes; Witt’s theorem (6) Symmetric and quasi-symmetric designs (7) The matrices of t-subsets versus k-subsets, or t-uniform hy- pergaphs (8) Null designs (trades) (9) A diagonal form for Nt (10) A zero-sum Ramsey-type problem (11) Diagonal forms for matrices arising from simple graphs 1. Smith and diagonal form Given an r by m integer matrix A, there exist unimodular matrices E and F ,ofordersr and m,sothatEAF = D where D is an r by m diagonal matrix. Here ‘diagonal’ means that the (i, j)-entry of D is 0 unless i = j; but D is not necessarily square. We call any matrix D that arises in this way a diagonal form for A. As a simple example, ⎛ ⎞ 0 −13 10 314⎜ ⎟ 100 ⎝1 −1 −1⎠ = . 21 4 −27 050 01−2 The matrix on the right is a diagonal form for the middle matrix on the left. Let the diagonal entries of D be d1,d2,d3,.... ⎛ ⎞ 20 0 ··· ⎜ ⎟ ⎜0240 ···⎟ ⎜ ⎟ ⎝0 0 120 ···⎠ . ... If all diagonal entries di are nonnegative and di divides di+1 for i =1, 2,...,thenD is called the integer Smith normal form of A, or simply the Smith form of A, and the integers di are called the invariant factors,ortheelementary divisors of A.TheSmith form is unique; the unimodular matrices E and F are not. -
Left Eigenvector of a Stochastic Matrix
Advances in Pure Mathematics, 2011, 1, 105-117 doi:10.4236/apm.2011.14023 Published Online July 2011 (http://www.SciRP.org/journal/apm) Left Eigenvector of a Stochastic Matrix Sylvain Lavalle´e Departement de mathematiques, Universite du Quebec a Montreal, Montreal, Canada E-mail: [email protected] Received January 7, 2011; revised June 7, 2011; accepted June 15, 2011 Abstract We determine the left eigenvector of a stochastic matrix M associated to the eigenvalue 1 in the commu- tative and the noncommutative cases. In the commutative case, we see that the eigenvector associated to the eigenvalue 0 is (,,NN1 n ), where Ni is the ith principal minor of NMI= n , where In is the 11 identity matrix of dimension n . In the noncommutative case, this eigenvector is (,P1 ,Pn ), where Pi is the sum in aij of the corresponding labels of nonempty paths starting from i and not passing through i in the complete directed graph associated to M . Keywords: Generic Stochastic Noncommutative Matrix, Commutative Matrix, Left Eigenvector Associated To The Eigenvalue 1, Skew Field, Automata 1. Introduction stochastic free field and that the vector 11 (,,PP1 n ) is fixed by our matrix; moreover, the sum 1 It is well known that 1 is one of the eigenvalue of a of the Pi is equal to 1, hence they form a kind of stochastic matrix (i.e. the sum of the elements of each noncommutative limiting probability. row is equal to 1) and its associated right eigenvector is These results have been proved in [1] but the proof the vector (1,1, ,1)T . -
Hadamard and Conference Matrices
Hadamard and conference matrices Peter J. Cameron December 2011 with input from Dennis Lin, Will Orrick and Gordon Royle Now det(H) is equal to the volume of the n-dimensional parallelepiped spanned by the rows of H. By assumption, each row has Euclidean length at most n1/2, so that det(H) ≤ nn/2; equality holds if and only if I every entry of H is ±1; > I the rows of H are orthogonal, that is, HH = nI. A matrix attaining the bound is a Hadamard matrix. Hadamard's theorem Let H be an n × n matrix, all of whose entries are at most 1 in modulus. How large can det(H) be? A matrix attaining the bound is a Hadamard matrix. Hadamard's theorem Let H be an n × n matrix, all of whose entries are at most 1 in modulus. How large can det(H) be? Now det(H) is equal to the volume of the n-dimensional parallelepiped spanned by the rows of H. By assumption, each row has Euclidean length at most n1/2, so that det(H) ≤ nn/2; equality holds if and only if I every entry of H is ±1; > I the rows of H are orthogonal, that is, HH = nI. Hadamard's theorem Let H be an n × n matrix, all of whose entries are at most 1 in modulus. How large can det(H) be? Now det(H) is equal to the volume of the n-dimensional parallelepiped spanned by the rows of H. By assumption, each row has Euclidean length at most n1/2, so that det(H) ≤ nn/2; equality holds if and only if I every entry of H is ±1; > I the rows of H are orthogonal, that is, HH = nI. -
One Method for Construction of Inverse Orthogonal Matrices
ONE METHOD FOR CONSTRUCTION OF INVERSE ORTHOGONAL MATRICES, P. DIÞÃ National Institute of Physics and Nuclear Engineering, P.O. Box MG6, Bucharest, Romania E-mail: [email protected] Received September 16, 2008 An analytical procedure to obtain parametrisation of complex inverse orthogonal matrices starting from complex inverse orthogonal conference matrices is provided. These matrices, which depend on nonzero complex parameters, generalize the complex Hadamard matrices and have applications in spin models and digital signal processing. When the free complex parameters take values on the unit circle they transform into complex Hadamard matrices. 1. INTRODUCTION In a seminal paper [1] Sylvester defined a general class of orthogonal matrices named inverse orthogonal matrices and provided the first examples of what are nowadays called (complex) Hadamard matrices. A particular class of inverse orthogonal matrices Aa()ij are those matrices whose inverse is given 1 t by Aa(1ij ) (1 a ji ) , where t means transpose, and their entries aij satisfy the relation 1 AA nIn (1) where In is the n-dimensional unit matrix. When the entries aij take values on the unit circle A–1 coincides with the Hermitian conjugate A* of A, and in this case (1) is the definition of complex Hadamard matrices. Complex Hadamard matrices have applications in quantum information theory, several branches of combinatorics, digital signal processing, etc. Complex orthogonal matrices unexpectedly appeared in the description of topological invariants of knots and links, see e.g. Ref. [2]. They were called two- weight spin models being related to symmetric statistical Potts models. These matrices have been generalized to two-weight spin models, also called , Paper presented at the National Conference of Physics, September 10–13, 2008, Bucharest-Mãgurele, Romania. -
Hadamard Matrices Include
Hadamard and conference matrices Peter J. Cameron University of St Andrews & Queen Mary University of London Mathematics Study Group with input from Rosemary Bailey, Katarzyna Filipiak, Joachim Kunert, Dennis Lin, Augustyn Markiewicz, Will Orrick, Gordon Royle and many happy returns . Happy Birthday, MSG!! Happy Birthday, MSG!! and many happy returns . Now det(H) is equal to the volume of the n-dimensional parallelepiped spanned by the rows of H. By assumption, each row has Euclidean length at most n1/2, so that det(H) ≤ nn/2; equality holds if and only if I every entry of H is ±1; > I the rows of H are orthogonal, that is, HH = nI. A matrix attaining the bound is a Hadamard matrix. This is a nice example of a continuous problem whose solution brings us into discrete mathematics. Hadamard's theorem Let H be an n × n matrix, all of whose entries are at most 1 in modulus. How large can det(H) be? A matrix attaining the bound is a Hadamard matrix. This is a nice example of a continuous problem whose solution brings us into discrete mathematics. Hadamard's theorem Let H be an n × n matrix, all of whose entries are at most 1 in modulus. How large can det(H) be? Now det(H) is equal to the volume of the n-dimensional parallelepiped spanned by the rows of H. By assumption, each row has Euclidean length at most n1/2, so that det(H) ≤ nn/2; equality holds if and only if I every entry of H is ±1; > I the rows of H are orthogonal, that is, HH = nI. -
Lattices from Tight Equiangular Frames Albrecht Böttcher
Claremont Colleges Scholarship @ Claremont CMC Faculty Publications and Research CMC Faculty Scholarship 1-1-2016 Lattices from tight equiangular frames Albrecht Böttcher Lenny Fukshansky Claremont McKenna College Stephan Ramon Garcia Pomona College Hiren Maharaj Pomona College Deanna Needell Claremont McKenna College Recommended Citation Albrecht Böttcher, Lenny Fukshansky, Stephan Ramon Garcia, Hiren Maharaj and Deanna Needell. “Lattices from equiangular tight frames.” Linear Algebra and its Applications, vol. 510, 395–420, 2016. This Article is brought to you for free and open access by the CMC Faculty Scholarship at Scholarship @ Claremont. It has been accepted for inclusion in CMC Faculty Publications and Research by an authorized administrator of Scholarship @ Claremont. For more information, please contact [email protected]. Lattices from tight equiangular frames Albrecht B¨ottcher, Lenny Fukshansky, Stephan Ramon Garcia, Hiren Maharaj, Deanna Needell Abstract. We consider the set of all linear combinations with integer coefficients of the vectors of a unit tight equiangular (k, n) frame and are interested in the question whether this set is a lattice, that is, a discrete additive subgroup of the k-dimensional Euclidean space. We show that this is not the case if the cosine of the angle of the frame is irrational. We also prove that the set is a lattice for n = k + 1 and that there are infinitely many k such that a lattice emerges for n = 2k. We dispose of all cases in dimensions k at most 9. In particular, we show that a (7,28) frame generates a strongly eutactic lattice and give an alternative proof of Roland Bacher’s recent observation that this lattice is perfect. -
SYMMETRIC CONFERENCE MATRICES of ORDER Pq2 + 1
Can. J. Math., Vol. XXX, No. 2, 1978, pp. 321-331 SYMMETRIC CONFERENCE MATRICES OF ORDER pq2 + 1 RUDOLF MATHON Introduction and definitions. A conference matrix of order n is a square matrix C with zeros on the diagonal and zbl elsewhere, which satisfies the orthogonality condition CCT = (n — 1)1. If in addition C is symmetric, C = CT, then its order n is congruent to 2 modulo 4 (see [5]). Symmetric conference matrices (C) are related to several important combinatorial configurations such as regular two-graphs, equiangular lines, Hadamard matrices and balanced incomplete block designs [1; 5; and 7, pp. 293-400]. We shall require several definitions. A strongly regular graph with parameters (v, k, X, /z) is an undirected regular graph of order v and degree k with adjacency matrix A = AT satisfying (1.1) A2 + 0* - \)A + (/x - k)I = \xJ, AJ = kJ where / is the all one matrix. Note that in a strongly regular graph any two adjacent (non-adjacent) vertices are adjacent to exactly X(/x) other vertices. An easy counting argument implies the following relation between the param eters v, k, X and /x; (1.2) k(k - X - 1) = (v - k - l)/x. A strongly regular graph is said to be pseudo-cyclic (PC) if v — 1 = 2k = 4/x. From (1.1) and (1.2) it is readily deduced that a PC-graph has parameters of the form (4/ + 1, 2/, t — 1, /), where t > 0 is an integer. We note that the complement of a PC-graph is again pseudo-cyclic with the same parameters as the original graph (though not necessarily isomorphic to it), i.e. -
ON EUCLIDEAN DISTANCE MATRICES of GRAPHS∗ 1. Introduction. a Matrix D ∈ R N×N Is a Euclidean Distance Matrix (EDM), If Ther
Electronic Journal of Linear Algebra ISSN 1081-3810 A publication of the International Linear Algebra Society Volume 26, pp. 574-589, August 2013 ELA ON EUCLIDEAN DISTANCE MATRICES OF GRAPHS∗ GASPERˇ JAKLICˇ † AND JOLANDA MODIC‡ Abstract. 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. Key words. Graph, Euclidean distance matrix, Distance, Eigenvalue. AMS subject classifications. 15A18, 05C50, 05C12. n n 1. Introduction. A matrix D R × is a Euclidean distance matrix (EDM), ∈ if there exist points x Rr, i =1, 2,...,n, such that d = x x 2. The minimal i ∈ ij k i − j k possible r is called an embedding dimension (see, e.g., [3]). Euclidean distance matri- ces were introduced by Menger in 1928, later they were studied by Schoenberg [13], and other authors. They have many interesting properties, and are used in various applications in linear algebra, graph theory, and bioinformatics. A natural problem is to study configurations of points xi, where only distances between them are known. n n Definition 1.1. A matrix D = (d ) R × is hollow, if d = 0 for all ij ∈ ii i =1, 2,...,n. There are various characterizations of EDMs. n n Lemma 1.2. [8, Lemma 5.3] Let a symmetric hollow matrix D R × have only ∈ one positive eigenvalue λ and the corresponding eigenvector e := [1, 1,..., 1]T Rn. -
A Submodular Approach
SPARSEST NETWORK SUPPORT ESTIMATION: A SUBMODULAR APPROACH Mario Coutino, Sundeep Prabhakar Chepuri, Geert Leus Delft University of Technology ABSTRACT recovery of the adjacency and normalized Laplacian matrix, in the In this work, we address the problem of identifying the underlying general case, due to the convex formulation of the recovery problem, network structure of data. Different from other approaches, which solutions that only retrieve the support approximately are obtained, are mainly based on convex relaxations of an integer problem, here i.e., thresholding operations are required. we take a distinct route relying on algebraic properties of a matrix Taking this issue into account, and following arguments closely representation of the network. By describing what we call possible related to the ones used for network topology estimation through ambiguities on the network topology, we proceed to employ sub- spectral templates [10], the main aim of this work is twofold: (i) to modular analysis techniques for retrieving the network support, i.e., provide a proper characterization of a particular set of matrices that network edges. To achieve this we only make use of the network are of high importance within graph signal processing: fixed-support modes derived from the data. Numerical examples showcase the ef- matrices with the same eigenbasis, and (ii) to provide a pure com- fectiveness of the proposed algorithm in recovering the support of binatorial algorithm, based on the submodular machinery that has sparse networks. been proven useful in subset selection problems within signal pro- cessing [18–22], as an alternative to traditional convex relaxations. Index Terms— Graph signal processing, graph learning, sparse graphs, network deconvolution, network topology inference. -
MATRIX THEORY and APPLICATIONS Edited by Charles R
http://dx.doi.org/10.1090/psapm/040 AMS SHORT COURSE LECTURE NOTES Introductory Survey Lectures A Subseries of Proceedings of Symposia in Applied Mathematics Volume 40 MATRIX THEORY AND APPLICATIONS Edited by Charles R. Johnson (Phoenix, Arizona, January 1989) Volume 39 CHAOS AND FRACTALS: THE MATHEMATICS BEHIND THE COMPUTER GRAPHICS Edited by Robert L.. Devaney and Linda Keen (Providence, Rhode Island, August 1988) Volume 38 COMPUTATIONAL COMPLEXITY THEORY Edited by Juris Hartmanis (Atlanta, Georgia, January 1988) Volume 37 MOMENTS IN MATHEMATICS Edited by Henry J. Landau (San Antonio, Texas, January 1987) Volume 36 APPROXIMATION THEORY Edited by Carl de Boor (New Orleans, Louisiana, January 1986) Volume 35 ACTUARIAL MATHEMATICS Edited by Harry H. Panjer (Laramie, Wyoming, August 1985) Volume 34 MATHEMATICS OF INFORMATION PROCESSING Edited by Michael Anshel and William Gewirtz (Louisville, Kentucky, January 1984) Volume 33 FAIR ALLOCATION Edited by H. Peyton Young (Anaheim, California, January 1985) Volume 32 ENVIRONMENTAL AND NATURAL RESOURCE MATHEMATICS Edited by R. W. McKelvey (Eugene, Oregon, August 1984) Volume 31 COMPUTER COMMUNICATIONS Edited by B. Gopinath (Denver, Colorado, January 1983) Volume 30 POPULATION BIOLOGY Edited by Simon A. Levin (Albany, New York, August 1983) Volume 29 APPLIED CRYPTOLOGY, CRYPTOGRAPHIC PROTOCOLS, AND COMPUTER SECURITY MODELS By R. A. DeMillo, G. I. Davida, D. P. Dobkin, M. A. Harrison, and R. J. Lipton (San Francisco, California, January 1981) Volume 28 STATISTICAL DATA ANALYSIS Edited by R. Gnanadesikan (Toronto, Ontario, August 1982) Volume 27 COMPUTED TOMOGRAPHY Edited by L. A. Shepp (Cincinnati, Ohio, January 1982) Volume 26 THE MATHEMATICS OF NETWORKS Edited by S. A. Burr (Pittsburgh, Pennsylvania, August 1981) Volume 25 OPERATIONS RESEARCH: MATHEMATICS AND MODELS Edited by S.