DOCSLIB.ORG

Eigenvalues of Euclidean Distance Matrices and Rs-Majorization on R2

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
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. Let Λ be the set of n n n × EDMs, and Ωn(e) be the set of n n positive semidefinite matrices B such that Be = 0. × 1 Then the linear mapping τ :Λ Ω (e) defined by τ(D) = − P DP is invertible, and its n → n 2 inverse mapping, say κ :Ω (e) Λ is given by κ(B) = bet + ebt 2B with b = diag(B), n → n − where diag(B) is the vector consisting of the diagonal elements of B. For general refrence on this topic see, e.g. [1]. Majorization is one of the vital topics in mathematics and statistics. It plays a basic role in matrix theory. One can see some type of majorization in [2]-[13]. In this paper, the relation between the eigenvalues of EDMs and those of the corresponding positive ∗Speaker 408 www.SID.ir 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.: 2{4 2 semidefinite matrices respect to rs on R will be investigated. An nonnegative matrix R ≺ is called row stochastic if the sum of entries of each row of R is equal to one. The following notation will be fixed throughout the paper. m m Co(A) := λiai m N, λi 0, λi = 1, ai A, i Nm , { i=1 | ∈ ≥ i=1 ∈ ∈ } for a subset A Rn; P ⊂ P Sgn α be 1 if α > 0 and be 1 if α < 0, Sgn 0 can be 1 or 1; { } − { } − [T ] be the matrix representation of a linear function T : Rn Rn with respect to the → standard basis; ri be the sum of entries on the ith row of [T ]. A linear function T : Rn Rn is said to be a linear preserver (strong linear preserver) −→ of if T (x) T (y) whenever x y (T (x) T (y) if and only if x y). ∼ ∼ ∼ ∼ ∼ 1.1 Rs-majorization n We introduce the relation rs on R and we state some properties of rs-majorization on ≺ R2. Definition 1.1. A matrix R M with nonnegative entries is called row stochastic if the ∈ n sum of entries of each row of R is equal to one. Definition 1.2. For two real vector x and y, we say that x is rs-majorized by y (denoted by x y) if there exists an n-by-n row stochastic matrix R with all its column entries ≺rs equal such that x = Ry. In this paper, we consider this relation on R2. The following proposition gives an equivalent condition for rs-majorization on R2. t t 2 Proposition 1.3. Let x = (x1, x2) , y = (y1, y2) R . Then x rs y if and only if ∈ ≺ x = x y , y . 1 2 ∈ C{ 1 2} 2 Here we state all (resp. strong) linear preservers of rs on R . ≺ 2 2 a b Theorem 1.4. Let T : R R be a linear function, and let [T ] = c d . Then T → preserves if and only if r = r , Sgn a = Sgn d = Sgn b = Sgn c . ≺rs 1 2 { } { } 6 { } { } 2 2 Theorem 1.5. A linear function T : R R strongly preserves rs if and only if → ≺ [T ] = αI for some α R 0 . ∈ \{ } 2 Main results Till the end of this section, the relation between the eigenvalues of EDMs and those of the 2 corresponding positive semidefinite matrices respect to rs on R will be specify. ≺ Theorem 2.1. Let B, B Ω (e), and let D = κ(B) and D = κ(B). Then ∈ 2 λ(B) λ(B) λ(D) λ(D) ≺rs ⇐⇒ ≺rs e e e e e 409 www.SID.ir 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.: 3{4 α α β β Proof. Since B, B Ω2(e), there exist α,β 0 such that B = α− α , B = β− β , and ∈ ≥ − − 0, 2α and 0, 2β are the set of eigenvalues of B and B, respectively. By the definition of { } 0{ 4α } 0 4β κ, D = ande D = . So 4α, 4α and 4β, 4β are thee set of eigenvalues 4α 0 4β 0 {− } {− } e of D and D, respectively. We see that λ(B) e λ(B) if and only if B = 0. Also, if λ(D) λ(D) if and only if ≺rs ≺rs D = 0. Hencee λ(B) λ(B) if and only if λ(D) λ(D). ≺rs ≺rs e e e e References [1] A. Y. Alfakih, On the eigenvalues of Euclidean distance matrices, Comput. Appl. Math., 27, 2008, 237-250. [2] T. Ando, Majorization, doubly stochastic matrices, and comparison of eigenval- ues, Linear Algebra Appl., 118, 1989, 163-248. [3] A. Armandnejad, Right gw-majorization on Mn,m, Bull. Iranian math. Soc., 35, 2009, no. 2, 69-76. [4] A. Armandnejad and H. Heydari, Linear preserving gd-majorization functions from Mn,m to Mn,k, Bull. Iranian math. Soc., 37, 2011, no. 1, 215-224. [5] A. Armandnejad and A. Ilkhanizadeh Manesh, Gut-majorization on Mn,m and its linear preservers, Electron. J. Linear Algebra, 23, 2012, 646-654. [6] A. Armandnejad and A. Salemi, On linear preservers of lgw-majorization on Mn,m, Bull. Malays. Math. Soc., 35, 2, 2012, no. 3, 755-764. [7] A. Armandnejad and A. Salemi, The structure of linear preservers of gs- majorization, Bull. Iranian Math. Soc., 32, 2006, no. 2, 31-42. [8] H. Chiang and C. K. Li, Generalized doubly stochastic matrices and linear preservers, Linear and Multilinear Algebra, 53, 2005, 1-11. [9] A. M. Hasani and M. Radjabalipour, On linear preservers of (right) matrix majoriza- tion, Linear Algebra Appl., 423, 2007, 255-261. [10] A. M. Hasani and M. Radjabalipour, The structure of linear operators strongly pre- serving majorizations of matrices, Electronic Journal of Linear Algebra, 15, 2006, 260-268. [11] A. Ilkhanizadeh Manesh, Linear functions preserving sut-majorization on Rn, Iranian Journal of Mathematical Sciences and Informatics, (submission). [12] A. Ilkhanizadeh Manesh, Right gut-Majorization on Mn,m, Electron. J. Linear Alge- bra, (submission). [13] A. Ilkhanizadeh Manesh and A. Armandnejad, Ut-Majorization on Rn and its Linear Preservers, Operator Theory: Advances and Applications, 242, 2014, 253-259. 410 www.SID.ir 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.: 4{4 Email: [email protected]; [email protected] Email: [email protected] 411 www.SID.ir .
Load more

Recommended publications
  • Clustering by Left-Stochastic Matrix Factorization
    Clustering by Left-Stochastic Matrix Factorization Raman Arora [email protected] Maya R. Gupta [email protected] Amol Kapila [email protected] Maryam Fazel [email protected] University of Washington, Seattle, WA 98103, USA Abstract 1.1. Related Work in Matrix Factorization Some clustering objective functions can be written as We propose clustering samples given their matrix factorization objectives. Let n feature vectors d×n pairwise similarities by factorizing the sim- be gathered into a feature-vector matrix X 2 R . T d×k ilarity matrix into the product of a clus- Consider the model X ≈ FG , where F 2 R can ter probability matrix and its transpose. be interpreted as a matrix with k cluster prototypes n×k We propose a rotation-based algorithm to as its columns, and G 2 R is all zeros except for compute this left-stochastic decomposition one (appropriately scaled) positive entry per row that (LSD). Theoretical results link the LSD clus- indicates the nearest cluster prototype. The k-means tering method to a soft kernel k-means clus- clustering objective follows this model with squared tering, give conditions for when the factor- error, and can be expressed as (Ding et al., 2005): ization and clustering are unique, and pro- T 2 arg min kX − FG kF ; (1) vide error bounds. Experimental results on F;GT G=I simulated and real similarity datasets show G≥0 that the proposed method reliably provides accurate clusterings. where k · kF is the Frobenius norm, and inequality G ≥ 0 is component-wise. This follows because the combined constraints G ≥ 0 and GT G = I force each row of G to have only one positive element.
    [Show full text]
  • 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.
    [Show full text]
  • Similarity-Based Clustering by Left-Stochastic Matrix Factorization
    JournalofMachineLearningResearch14(2013)1715-1746 Submitted 1/12; Revised 11/12; Published 7/13 Similarity-based Clustering by Left-Stochastic Matrix Factorization Raman Arora [email protected] Toyota Technological Institute 6045 S. Kenwood Ave Chicago, IL 60637, USA Maya R. Gupta [email protected] Google 1225 Charleston Rd Mountain View, CA 94301, USA Amol Kapila [email protected] Maryam Fazel [email protected] Department of Electrical Engineering University of Washington Seattle, WA 98195, USA Editor: Inderjit Dhillon Abstract For similarity-based clustering, we propose modeling the entries of a given similarity matrix as the inner products of the unknown cluster probabilities. To estimate the cluster probabilities from the given similarity matrix, we introduce a left-stochastic non-negative matrix factorization problem. A rotation-based algorithm is proposed for the matrix factorization. Conditions for unique matrix factorizations and clusterings are given, and an error bound is provided. The algorithm is partic- ularly efficient for the case of two clusters, which motivates a hierarchical variant for cases where the number of desired clusters is large. Experiments show that the proposed left-stochastic decom- position clustering model produces relatively high within-cluster similarity on most data sets and can match given class labels, and that the efficient hierarchical variant performs surprisingly well. Keywords: clustering, non-negative matrix factorization, rotation, indefinite kernel, similarity, completely positive 1. Introduction Clustering is important in a broad range of applications, from segmenting customers for more ef- fective advertising, to building codebooks for data compression. Many clustering methods can be interpreted in terms of a matrix factorization problem.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • Alternating Sign Matrices and Polynomiography
    Alternating Sign Matrices and Polynomiography Bahman Kalantari Department of Computer Science Rutgers University, USA [email protected] Submitted: Apr 10, 2011; Accepted: Oct 15, 2011; Published: Oct 31, 2011 Mathematics Subject Classifications: 00A66, 15B35, 15B51, 30C15 Dedicated to Doron Zeilberger on the occasion of his sixtieth birthday Abstract To each permutation matrix we associate a complex permutation polynomial with roots at lattice points corresponding to the position of the ones. More generally, to an alternating sign matrix (ASM) we associate a complex alternating sign polynomial. On the one hand visualization of these polynomials through polynomiography, in a combinatorial fashion, provides for a rich source of algo- rithmic art-making, interdisciplinary teaching, and even leads to games. On the other hand, this combines a variety of concepts such as symmetry, counting and combinatorics, iteration functions and dynamical systems, giving rise to a source of research topics. More generally, we assign classes of polynomials to matrices in the Birkhoff and ASM polytopes. From the characterization of vertices of these polytopes, and by proving a symmetry-preserving property, we argue that polynomiography of ASMs form building blocks for approximate polynomiography for polynomials corresponding to any given member of these polytopes. To this end we offer an algorithm to express any member of the ASM polytope as a convex of combination of ASMs. In particular, we can give exact or approximate polynomiography for any Latin Square or Sudoku solution. We exhibit some images. Keywords: Alternating Sign Matrices, Polynomial Roots, Newton’s Method, Voronoi Diagram, Doubly Stochastic Matrices, Latin Squares, Linear Programming, Polynomiography 1 Introduction Polynomials are undoubtedly one of the most significant objects in all of mathematics and the sciences, particularly in combinatorics.
    [Show full text]
  • 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.
    [Show full text]
  • Representations of Stochastic Matrices
    Rotational (and Other) Representations of Stochastic Matrices Steve Alpern1 and V. S. Prasad2 1Department of Mathematics, London School of Economics, London WC2A 2AE, United Kingdom. email: [email protected] 2Department of Mathematics, University of Massachusetts Lowell, Lowell, MA. email: [email protected] May 27, 2005 Abstract Joel E. Cohen (1981) conjectured that any stochastic matrix P = pi;j could be represented by some circle rotation f in the following sense: Forf someg par- tition Si of the circle into sets consisting of …nite unions of arcs, we have (*) f g pi;j = (f (Si) Sj) = (Si), where denotes arc length. In this paper we show how cycle decomposition\ techniques originally used (Alpern, 1983) to establish Cohen’sconjecture can be extended to give a short simple proof of the Coding Theorem, that any mixing (that is, P N > 0 for some N) stochastic matrix P can be represented (in the sense of * but with Si merely measurable) by any aperiodic measure preserving bijection (automorphism) of a Lesbesgue proba- bility space. Representations by pointwise and setwise periodic automorphisms are also established. While this paper is largely expository, all the proofs, and some of the results, are new. Keywords: rotational representation, stochastic matrix, cycle decomposition MSC 2000 subject classi…cations. Primary: 60J10. Secondary: 15A51 1 Introduction An automorphism of a Lebesgue probability space (X; ; ) is a bimeasurable n bijection f : X X which preserves the measure : If S = Si is a non- ! f gi=1 trivial (all (Si) > 0) measurable partition of X; we can generate a stochastic n matrix P = pi;j by the de…nition f gi;j=1 (f (Si) Sj) pi;j = \ ; i; j = 1; : : : ; n: (1) (Si) Since the partition S is non-trivial, the matrix P has a positive invariant (stationary) distribution v = (v1; : : : ; vn) = ( (S1) ; : : : ; (Sn)) ; and hence (by de…nition) is recurrent.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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 .
    [Show full text]