DOCSLIB.ORG

Graph Spectra and Continuous Quantum Walks Gabriel Coutinho

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Graph Spectra and Continuous Quantum Walks Gabriel Coutinho Graph Spectra and Continuous Quantum Walks Gabriel Coutinho, Chris Godsil September 1, 2020 Preface If A is the adjacency matrix of a graph X, we define a transition matrix U(t) by U(t) = exp(itA). Physicists say that U(t) determines a continuous quantum walk. Most ques- tions they ask about these matrices concern the squared absolute values of the entries of U(t) (because these may be determined by measurement, and the entries themselves cannot be). The matrices U(t) are unitary and U(t) = U(−t), so we may say that the physicists are concerned with ques- tions about the entries of the Schur product M(t) = U(t) ◦ U(t) = U(t) ◦ U(−t). Since U(t) is unitary, the matrix M(t) is doubly stochastic. Hence each column determines a probability density on V (X) and there are two extreme cases: (a) A row of M(t) has one nonzero entry (necessarily equal to 1). (b) All entries in a row of U(t) are equal to 1/|V (X)|. In case (a), there are vertices a and b of X such that |U(t)a,b| = 1. If a =6 b, we have perfect state transfer from a to b at time t, otherwise we say that X is periodic at a at time t. Physicists are particularly interested in perfect state transfer. In case (b), if the entries of the a-row of M(t) is constant, we have local uniform mixing at a at time t. (Somewhat surprisingly these two concepts are connected—in a number of cases, perfect state transfer and local uniform mixing occur on the same graph.) The quest for graphs admitting perfect state transfer or uniform mixing has unveiled new results in spectral graph theory, and rich connections to other fields of mathematics. This also brought us to explore more topics in iii quantum information and their interplay with combinatorics. This book is, in part, a progress report on these questions. For us, the biggest surprise is the extent to which tools from algebraic graph theory prove useful. So we treat this in somewhat more detail than is strictly necessary. Some of it is standard, some is old stuff repackaged, and some is new material (e.g., controllability, strongly cospectral vertices) that has been developed to deal with quantum walks. But combinatorics is not everything: we also meet with Lie groups, various flavours of number theory, and almost periodic functions. (And so a second surprise is the number of different mathematical areas entangled with our topic.) we do not treat discrete quantum walks. We do not treat quantum algo- rithms or quantum computation, nor do we deal with questions about com- plexity, error correction, non-local games, and the quantum circuit model. We discuss a little of the related physics. We have focussed on questions that are both mathematically interesting and have some physical signifi- cance, since this overlap is often a sign of a fruitful outcome. We have had useful comments on these notes from many people, includ- ing Dave Witte Morris, Tino Tamon, Sasha Jurišic and members of his seminar, Alexis Hunt, David Feder, Henry Liu, Harmony Zhan, Nicholas Lai, Xiaohong Zhang. This is a preliminary version intended to be used a class notes for a grad course at UFMG in 2020. We are still working on it, and several chapters lack references, exercises, and revision. We wel- come any comments to [email protected] or [email protected]. Contents Preface iii Contents v I Introduction 1 1 Continuous Quantum Walks 3 1.1 Basics . 4 1.2 Products . 6 1.3 State Transfer and Mixing . 7 1.4 Symmetry and Periodicity . 8 1.5 Spectral Decomposition for Adjacency Matrices . 9 1.6 Using Spectral Decomposition . 11 1.7 Complete Graphs . 12 1.8 Bipartite Graphs . 13 1.9 Uniform Mixing on Bipartite Graphs and Hadamard matrices 15 1.10 Vertex-Transitive Graphs . 16 1.11 Pretty Good State Transfer . 18 1.12 Oriented Graphs . 20 1.13 Universal State Transfer . 21 1.14 Path Plots . 23 1.15 One More Plot . 27 2 Physics 29 2.1 Quantum Systems . 29 2.2 Density Matrices and POVMs . 31 2.3 Qubits and 2 × 2 Hermitians . 34 v Contents 2.4 Entanglement . 36 2.5 Walks: Adjacency Matrix . 37 2.6 Walks: More Matrices . 39 2.7 Orbits of Density Matrices . 40 II Spectra of Graphs 45 3 Equitable Partitions 47 3.1 Partitions and Projections . 47 3.2 Walks and orthogonal matrices . 50 3.3 Automorphisms . 52 3.4 Quotients of the d-cube and of complete bipartite graphs . 54 4 Spectrum and Walks 57 4.1 Formal power series . 58 4.2 Walk generating function . 59 4.3 Closed Walks at a Vertex . 60 4.4 Walks between two vertices . 62 4.5 The Jacobi trick . 64 4.6 All Walks . 65 4.7 1-Sums and 2-Sums . 68 4.8 Reduced Walks . 70 4.9 Hermitian matrices . 71 4.10 No transfer at all on some signed graphs . 72 5 Walk Modules 75 5.1 Walk Modules . 75 5.2 Dual degree and covering radius . 76 5.3 Controllable pairs, symmetries, rational functions . 78 5.4 Controllable subsets . 80 5.5 Controllable Graphs . 82 5.6 Isomorphism . 83 5.7 Isomorphism of Controllable Pairs . 86 5.8 Laplacians . 87 5.9 Control Theory . 88 6 Cospectral Vertices 93 vi Contents 6.1 Walk modules of cospectral vertices . 93 6.2 Characterizing Cospectral Vertices . 94 6.3 Walk-Regular Graphs . 95 6.4 State Transfer on Walk-Regular Graphs . 97 6.5 Parallel Vertices . 98 6.6 Strongly Cospectral Vertices . 99 6.7 Examples of Strongly Cospectral Vertices . 101 6.8 Characterizing Strongly Cospectral Vertices . 103 6.9 Matrix Algebras . 105 IIIState Transfer and Periodicity 109 7 State transfer 111 7.1 Three Inequalities . 112 7.2 Near Enough: Cospectrality . 114 7.3 Ratio Condition . 117 7.4 Using Integrality . 122 7.5 Perfect state transfer . 123 7.6 Bipartite and Regular Graphs . 126 7.7 No Control . 127 8 Recurrence and approximations 131 8.1 Periods . 131 8.2 Tori . 133 8.3 Dense Subgroups of U(n) . 137 8.4 Periods of Functions . 139 8.5 Almost Periodic . 140 8.6 Means . 141 8.7 Zeros of Transition Functions . 142 9 Real State Transfer 145 9.1 Real State Transfer . 146 9.2 Pretty Good State Transfer . 148 9.3 Algebras . 149 9.4 Timing . 150 9.5 Periodicity and Eigenvalues . 151 9.6 Algebraic States . 153 vii Contents 9.7 Fractional Revival . 155 9.8 Symmetry . 156 9.9 Commutativity . 158 9.10 Cospectrality . 163 9.11 Characterizing Fractional Revival . 164 10 Paths and Trees 169 10.1 Recurrences for Paths . 169 10.2 Eigenvalues and Eigenvectors . 171 10.3 Line Graphs and Laplacians . 175 10.4 Inverses . 177 10.5 Eigenthings for Laplacians of Paths . 178 10.6 No Transfer on Paths . 180 10.7 PST on Laplacians . 180 10.8 Twins . 182 10.9 No Laplacian Perfect State Transfer on Trees . 183 11 Pretty Good State Transfer 185 11.1 PGST on Paths . 185 11.2 Phases and Pretty Good State Transfer . 186 11.3 No PGST on Paths . 188 11.4 Path Laplacians . 190 11.5 Double Stars . 193 12 Joins and Products 197 12.1 Eigenvalues and Eigenvectors of Joins . 197 12.2 Spectral Idempotents for Joins . 199 12.3 The Transition Matrix of a Join . 201 12.4 Joins with K2 and K2 . 202 12.5 Irrational Periods and Phases . 205 12.6 The Direct Product . 206 12.7 Double Covers and Switching Graphs . 208 12.8 Laplacians and Complements . 210 IVAlgebraic Connections 213 13 Average Mixing 215 viii Contents 13.1 Average Mixing . 215 13.2 Properties of the Average Mixing Matrix . 217 13.3 Uniform Average Mixing? . 219 13.4 An Ordering . 220 13.5 Strongly Cospectral Vertices . 221 13.6 Integrality . 222 13.7 Near Enough Implies Strongly Cospectral . 224 13.8 Average Mixing on Paths . 225 13.9 Path Laplacians . 226 13.10 Cycles . 228 13.11 Pseudocyclic Schemes . 230 13.12 Average Mixing on Pseudocyclic Graphs . 232 13.13 Average Mixing on Oriented Graphs . 233 13.14 Average States . 234 14 Orthogonal Polynomials 237 14.1 Examples . 238 14.2 The Three-Term Recurrence . ..
Load more

Recommended publications
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • On Matrices with a Doubly Stochastic Pattern
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector JOURNAL OF MATHEMATICALANALYSISAND APPLICATIONS 34,648-652(1971) On Matrices with a Doubly Stochastic Pattern DAVID LONDON Technion-Israel Institute of Technology, Haifa Submitted by Ky Fan 1. INTRODUCTION In [7] Sinkhorn proved that if A is a positive square matrix, then there exist two diagonal matrices D, = {@r,..., d$) and D, = {dj2),..., di2r) with positive entries such that D,AD, is doubly stochastic. This problem was studied also by Marcus and Newman [3], Maxfield and Mint [4] and Menon [5]. Later Sinkhorn and Knopp [8] considered the same problem for A non- negative. Using a limit process of alternately normalizing the rows and columns sums of A, they obtained a necessary and sufficient condition for the existence of D, and D, such that D,AD, is doubly stochastic. Brualdi, Parter and Schneider [l] obtained the same theorem by a quite different method using spectral properties of some nonlinear operators. In this note we give a new proof of the same theorem. We introduce an extremal problem, and from the existence of a solution to this problem we derive the existence of D, and D, . This method yields also a variational characterization for JJTC1(din dj2’), which can be applied to obtain bounds for this quantity. We note that bounds for I7y-r (d,!‘) dj”)) may be of interest in connection with inequalities for the permanent of doubly stochastic matrices [3]. 2. PRELIMINARIES Let A = (aij) be an 12x n nonnegative matrix; that is, aij > 0 for i,j = 1,***, n.
    [Show full text]
  • An Inequality for Doubly Stochastic Matrices*
    JOURNAL OF RESEARCH of the National Bureau of Standards-B. Mathematical Sciences Vol. 80B, No.4, October-December 1976 An Inequality for Doubly Stochastic Matrices* Charles R. Johnson** and R. Bruce Kellogg** Institute for Basic Standards, National Bureau of Standards, Washington, D.C. 20234 (June 3D, 1976) Interrelated inequalities involving doubly stochastic matrices are presented. For example, if B is an n by n doubly stochasti c matrix, x any nonnegative vector and y = Bx, the n XIX,· •• ,x" :0:::; YIY" •• y ... Also, if A is an n by n nonnegotive matrix and D and E are positive diagonal matrices such that B = DAE is doubly stochasti c, the n det DE ;:::: p(A) ... , where p (A) is the Perron· Frobenius eigenvalue of A. The relationship between these two inequalities is exhibited. Key words: Diagonal scaling; doubly stochasti c matrix; P erron·Frobenius eigenvalue. n An n by n entry·wise nonnegative matrix B = (b i;) is called row (column) stochastic if l bi ; = 1 ;= 1 for all i = 1,. ',n (~l bij = 1 for all j = 1,' . ',n ). If B is simultaneously row and column stochastic then B is said to be doubly stochastic. We shall denote the Perron·Frobenius (maximal) eigenvalue of an arbitrary n by n entry·wise nonnegative matrix A by p (A). Of course, if A is stochastic, p (A) = 1. It is known precisely which n by n nonnegative matrices may be diagonally scaled by positive diagonal matrices D, E so that (1) B=DAE is doubly stochastic. If there is such a pair D, E, we shall say that A has property (").
    [Show full text]
  • Doubly Stochastic Normalization for Spectral Clustering
    Doubly Stochastic Normalization for Spectral Clustering Ron Zass and Amnon Shashua ∗ Abstract In this paper we focus on the issue of normalization of the affinity matrix in spec- tral clustering. We show that the difference between N-cuts and Ratio-cuts is in the error measure being used (relative-entropy versus L1 norm) in finding the closest doubly-stochastic matrix to the input affinity matrix. We then develop a scheme for finding the optimal, under Frobenius norm, doubly-stochastic approxi- mation using Von-Neumann’s successive projections lemma. The new normaliza- tion scheme is simple and efficient and provides superior clustering performance over many of the standardized tests. 1 Introduction The problem of partitioning data points into a number of distinct sets, known as the clustering problem, is central in data analysis and machine learning. Typically, a graph-theoretic approach to clustering starts with a measure of pairwise affinity Kij measuring the degree of similarity between points xi, xj, followed by a normalization step, followed by the extraction of the leading eigenvectors which form an embedded coordinate system from which the partitioning is readily available. In this domain there are three principle dimensions which make a successful clustering: (i) the affinity measure, (ii) the normalization of the affinity matrix, and (iii) the particular clustering algorithm. Common practice indicates that the former two are largely responsible for the performance whereas the particulars of the clustering process itself have a relatively smaller impact on the performance. In this paper we focus on the normalization of the affinity matrix. We first show that the existing popular methods Ratio-cut (cf.
    [Show full text]
  • Majorization Results for Zeros of Orthogonal Polynomials
    Majorization results for zeros of orthogonal polynomials ∗ Walter Van Assche KU Leuven September 4, 2018 Abstract We show that the zeros of consecutive orthogonal polynomials pn and pn−1 are linearly connected by a doubly stochastic matrix for which the entries are explicitly computed in terms of Christoffel numbers. We give similar results for the zeros of pn (1) and the associated polynomial pn−1 and for the zeros of the polynomial obtained by deleting the kth row and column (1 ≤ k ≤ n) in the corresponding Jacobi matrix. 1 Introduction 1.1 Majorization Recently S.M. Malamud [3, 4] and R. Pereira [6] have given a nice extension of the Gauss- Lucas theorem that the zeros of the derivative p′ of a polynomial p lie in the convex hull of the roots of p. They both used the theory of majorization of sequences to show that if x =(x1, x2,...,xn) are the zeros of a polynomial p of degree n, and y =(y1,...,yn−1) are the zeros of its derivative p′, then there exists a doubly stochastic (n − 1) × n matrix S such that y = Sx ([4, Thm. 4.7], [6, Thm. 5.4]). A rectangular (n − 1) × n matrix S =(si,j) is said to be doubly stochastic if si,j ≥ 0 and arXiv:1607.03251v1 [math.CA] 12 Jul 2016 n n− 1 n − 1 s =1, s = . i,j i,j n j=1 i=1 X X In this paper we will use the notion of majorization to rephrase some known results for the zeros of orthogonal polynomials on the real line.
    [Show full text]
  • The Symbiotic Relationship of Combinatorics and Matrix Theory
    The Symbiotic Relationship of Combinatorics and Matrix Theory Richard A. Brualdi* Department of Mathematics University of Wisconsin Madison. Wisconsin 53 706 Submitted by F. Uhlig ABSTRACT This article demonstrates the mutually beneficial relationship that exists between combinatorics and matrix theory. 3 1. INTRODUCTION According to The Random House College Dictionary (revised edition, 1984) the word symbiosis is defined as follows: symbiosis: the living together of two dissimilar organisms, esp. when this association is mutually beneficial. In applying, as I am, this definition to the relationship between combinatorics and matrix theory, I would have preferred that the qualifying word “dissimilar” was omitted. Are combinatorics and matrix theory really dissimilar subjects? On the other hand I very much like the inclusion of the phrase “when the association is mutually beneficial” because I believe the development of each of these subjects shows that combinatorics and matrix theory have a mutually beneficial relationship. The fruits of this mutually beneficial relationship is the subject that I call combinatorial matrix theory or, for brevity here, CMT. Viewed broadly, as I do, CMT is an amazingly rich and diverse subject. If it is * Research partially supported by National Science Foundation grant DMS-8901445 and National Security Agency grant MDA904-89 H-2060. LZNEAR ALGEBRA AND ITS APPLZCATZONS 162-164:65-105 (1992) 65 0 Elsevier Science Publishing Co., Inc., 1992 655 Avenue of the Americas, New York, NY 10010 0024.3795/92/$5.00 66 RICHARD A. BRUALDI combinatorial and uses matrix theory r in its formulation or proof, it’s CMT. If it is matrix theory and has a combinatorial component to it, it’s CMT.
    [Show full text]