Smith Normal Forms of Incidence Matrices
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Quiz7 Problem 1
Quiz 7 Quiz7 Problem 1. Independence The Problem. In the parts below, cite which tests apply to decide on independence or dependence. Choose one test and show complete details. 1 ! 1 ! (a) Vectors ~v = ;~v = 1 −1 2 1 0 1 1 0 1 1 0 1 1 B C B C B C (b) Vectors ~v1 = @ −1 A ;~v2 = @ 1 A ;~v3 = @ 0 A 0 −1 −1 2 5 (c) Vectors ~v1;~v2;~v3 are data packages constructed from the equations y = x ; y = x ; y = x10 on (−∞; 1). (d) Vectors ~v1;~v2;~v3 are data packages constructed from the equations y = 1 + x; y = 1 − x; y = x3 on (−∞; 1). Basic Test. To show three vectors are independent, form the system of equations c1~v1 + c2~v2 + c3~v3 = ~0; then solve for c1; c2; c3. If the only solution possible is c1 = c2 = c3 = 0, then the vectors are independent. Linear Combination Test. A list of vectors is independent if and only if each vector in the list is not a linear combination of the remaining vectors, and each vector in the list is not zero. Subset Test. Any nonvoid subset of an independent set is independent. The basic test leads to three quick independence tests for column vectors. All tests use the augmented matrix A of the vectors, which for 3 vectors is A =< ~v1j~v2j~v3 >. Rank Test. The vectors are independent if and only if rank(A) equals the number of vectors. Determinant Test. Assume A is square. The vectors are independent if and only if the deter- minant of A is nonzero. -
Adjacency and Incidence Matrices
Adjacency and Incidence Matrices 1 / 10 The Incidence Matrix of a Graph Definition Let G = (V ; E) be a graph where V = f1; 2;:::; ng and E = fe1; e2;:::; emg. The incidence matrix of G is an n × m matrix B = (bik ), where each row corresponds to a vertex and each column corresponds to an edge such that if ek is an edge between i and j, then all elements of column k are 0 except bik = bjk = 1. 1 2 e 21 1 13 f 61 0 07 3 B = 6 7 g 40 1 05 4 0 0 1 2 / 10 The First Theorem of Graph Theory Theorem If G is a multigraph with no loops and m edges, the sum of the degrees of all the vertices of G is 2m. Corollary The number of odd vertices in a loopless multigraph is even. 3 / 10 Linear Algebra and Incidence Matrices of Graphs Recall that the rank of a matrix is the dimension of its row space. Proposition Let G be a connected graph with n vertices and let B be the incidence matrix of G. Then the rank of B is n − 1 if G is bipartite and n otherwise. Example 1 2 e 21 1 13 f 61 0 07 3 B = 6 7 g 40 1 05 4 0 0 1 4 / 10 Linear Algebra and Incidence Matrices of Graphs Recall that the rank of a matrix is the dimension of its row space. Proposition Let G be a connected graph with n vertices and let B be the incidence matrix of G. -
A Brief Introduction to Spectral Graph Theory
A BRIEF INTRODUCTION TO SPECTRAL GRAPH THEORY CATHERINE BABECKI, KEVIN LIU, AND OMID SADEGHI MATH 563, SPRING 2020 Abstract. There are several matrices that can be associated to a graph. Spectral graph theory is the study of the spectrum, or set of eigenvalues, of these matrices and its relation to properties of the graph. We introduce the primary matrices associated with graphs, and discuss some interesting questions that spectral graph theory can answer. We also discuss a few applications. 1. Introduction and Definitions This work is primarily based on [1]. We expect the reader is familiar with some basic graph theory and linear algebra. We begin with some preliminary definitions. Definition 1. Let Γ be a graph without multiple edges. The adjacency matrix of Γ is the matrix A indexed by V (Γ), where Axy = 1 when there is an edge from x to y, and Axy = 0 otherwise. This can be generalized to multigraphs, where Axy becomes the number of edges from x to y. Definition 2. Let Γ be an undirected graph without loops. The incidence matrix of Γ is the matrix M, with rows indexed by vertices and columns indexed by edges, where Mxe = 1 whenever vertex x is an endpoint of edge e. For a directed graph without loss, the directed incidence matrix N is defined by Nxe = −1; 1; 0 corresponding to when x is the head of e, tail of e, or not on e. Definition 3. Let Γ be an undirected graph without loops. The Laplace matrix of Γ is the matrix L indexed by V (G) with zero row sums, where Lxy = −Axy for x 6= y. -
Arxiv:1302.2015V2 [Cs.CG]
MATHEMATICS OF COMPUTATION Volume 00, Number 0, Pages 000–000 S 0025-5718(XX)0000-0 PERSISTENCE MODULES: ALGEBRA AND ALGORITHMS PRIMOZ SKRABA Joˇzef Stefan Institute, Ljubljana, Slovenia Artificial Intelligence Laboratory, Joˇzef Stefan Institute, Jamova 39, 1000 Ljubljana, Slovenia MIKAEL VEJDEMO-JOHANSSON Corresponding author Formerly: School of Computer Science, University of St Andrews, Scotland Computer Vision and Active Perception Lab, KTH, Teknikringen 14, 100 44 Stockholm, Sweden Abstract. Persistent homology was shown by Zomorodian and Carlsson [35] to be homology of graded chain complexes with coefficients in the graded ring k[t]. As such, the behavior of persistence modules — graded modules over k[t] — is an important part in the analysis and computation of persistent homology. In this paper we present a number of facts about persistence modules; ranging from the well-known but under-utilized to the reconstruction of techniques to work in a purely algebraic approach to persistent homology. In particular, the results we present give concrete algorithms to compute the persistent homology of a simplicial complex with torsion in the chain complex. Contents 1. Introduction 2 2. Persistent (Co-)Homology 3 arXiv:1302.2015v2 [cs.CG] 15 Feb 2013 3. Overview of Commutative Algebra 4 4. Applications 21 5. Conclusions & Future Work 23 References 23 Appendix A. Algorithm for Computing Presentations of Persistence Modules 25 Appendix B. Relative Persistent Homology Example 26 E-mail addresses: [email protected], [email protected]. 2010 Mathematics Subject Classification. 13P10, 55N35. c XXXX American Mathematical Society 1 2 PERSISTENCE MODULES: ALGEBRA AND ALGORITHMS 1. Introduction The ideas of topological persistence [20] and persistent homology [35] have had a fundamental impact on computational geometry and the newly spawned field of applied topology. -
Incidence Matrices and Interval Graphs
Pacific Journal of Mathematics INCIDENCE MATRICES AND INTERVAL GRAPHS DELBERT RAY FULKERSON AND OLIVER GROSS Vol. 15, No. 3 November 1965 PACIFIC JOURNAL OF MATHEMATICS Vol. 15, No. 3, 1965 INCIDENCE MATRICES AND INTERVAL GRAPHS D. R. FULKERSON AND 0. A. GROSS According to present genetic theory, the fine structure of genes consists of linearly ordered elements. A mutant gene is obtained by alteration of some connected portion of this structure. By examining data obtained from suitable experi- ments, it can be determined whether or not the blemished portions of two mutant genes intersect or not, and thus inter- section data for a large number of mutants can be represented as an undirected graph. If this graph is an "interval graph," then the observed data is consistent with a linear model of the gene. The problem of determining when a graph is an interval graph is a special case of the following problem concerning (0, l)-matrices: When can the rows of such a matrix be per- muted so as to make the l's in each column appear consecu- tively? A complete theory is obtained for this latter problem, culminating in a decomposition theorem which leads to a rapid algorithm for deciding the question, and for constructing the desired permutation when one exists. Let A — (dij) be an m by n matrix whose entries ai3 are all either 0 or 1. The matrix A may be regarded as the incidence matrix of elements el9 e2, , em vs. sets Sl9 S2, , Sn; that is, ai3 = 0 or 1 ac- cording as et is not or is a member of S3 . -
On the Theory of Point Weight Designs Royal Holloway
On the theory of Point Weight Designs Alexander W. Dent Technical Report RHUL–MA–2003–2 26 March 2001 Royal Holloway University of London Department of Mathematics Royal Holloway, University of London Egham, Surrey TW20 0EX, England http://www.rhul.ac.uk/mathematics/techreports Abstract A point-weight incidence structure is a structure of blocks and points where each point is associated with a positive integer weight. A point-weight design is a point-weight incidence structure where the sum of the weights of the points on a block is constant and there exist some condition that specifies the number of blocks that certain sets of points lie on. These structures share many similarities to classical designs. Chapter one provides an introduction to design theory and to some of the existing theory of point-weight designs. Chapter two develops a new type of point-weight design, termed a row-sum point-weight design, that has some of the matrix properties of classical designs. We examine the combinatorial aspects of these designs and show that a Fisher inequality holds and that this is dependent on certain combinatorial properties of the points of minimal weight. We define these points, and the designs containing them, to be either ‘awkward’ or ‘difficult’ depending on these properties. Chapter three extends the combinatorial analysis of row-sum point-weight designs. We examine structures that are simultaneously row-sum and point-sum point-weight designs, paying particular attention to the question of regularity. We also present several general construction techniques and specific examples of row-sum point-weight designs that are generated using these techniques. -
Second Edition Volume I
Second Edition Thomas Beth Universitat¨ Karlsruhe Dieter Jungnickel Universitat¨ Augsburg Hanfried Lenz Freie Universitat¨ Berlin Volume I PUBLISHED BY THE PRESS SYNDICATE OF THE UNIVERSITY OF CAMBRIDGE The Pitt Building, Trumpington Street, Cambridge, United Kingdom CAMBRIDGE UNIVERSITY PRESS The Edinburgh Building, Cambridge CB2 2RU, UK www.cup.cam.ac.uk 40 West 20th Street, New York, NY 10011-4211, USA www.cup.org 10 Stamford Road, Oakleigh, Melbourne 3166, Australia Ruiz de Alarc´on 13, 28014 Madrid, Spain First edition c Bibliographisches Institut, Zurich, 1985 c Cambridge University Press, 1993 Second edition c Cambridge University Press, 1999 This book is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 1999 Printed in the United Kingdom at the University Press, Cambridge Typeset in Times Roman 10/13pt. in LATEX2ε[TB] A catalogue record for this book is available from the British Library Library of Congress Cataloguing in Publication data Beth, Thomas, 1949– Design theory / Thomas Beth, Dieter Jungnickel, Hanfried Lenz. – 2nd ed. p. cm. Includes bibliographical references and index. ISBN 0 521 44432 2 (hardbound) 1. Combinatorial designs and configurations. I. Jungnickel, D. (Dieter), 1952– . II. Lenz, Hanfried. III. Title. QA166.25.B47 1999 5110.6 – dc21 98-29508 CIP ISBN 0 521 44432 2 hardback Contents I. Examples and basic definitions .................... 1 §1. Incidence structures and incidence matrices ............ 1 §2. Block designs and examples from affine and projective geometry ........................6 §3. t-designs, Steiner systems and configurations ......... -
Invariants of Incidence Matrices
Invariants of Incidence Matrices Chris Godsil (University of Waterloo), Peter Sin (University of Florida), Qing Xiang (University of Delaware). March 29 – April 3, 2009 1 Overview of the Field Incidence matrices arise whenever one attempts to find invariants of a relation between two (usually finite) sets. Researchers in design theory, coding theory, algebraic graph theory, representation theory, and finite geometry all encounter problems about modular ranks and Smith normal forms (SNF) of incidence matrices. For example, the work by Hamada [9] on the dimension of the code generated by r-flats in a projective geometry was motivated by problems in coding theory (Reed-Muller codes) and finite geometry; the work of Wilson [18] on the diagonal forms of subset-inclusion matrices was motivated by questions on existence of designs; and the papers [2] and [5] on p-ranks and Smith normal forms of subspace-inclusion matrices have their roots in representation theory and finite geometry. An impression of the current directions in research can be gained by considering our level of understanding of some fundamental examples. Incidence of subsets of a finite set Let Xr denote the set of subsets of size r in a finite set X. We can consider various incidence relations between Xr and Xs, such as inclusion, empty intersection or, more generally, intersection of fixed size t. These incidence systems are of central importance in the theory of designs, where they play a key role in Wilson’s fundamental work on existence theorems. They also appear in the theory of association schemes. 1 2 Incidence of subspaces of a finite vector space This class of incidence systems is the exact q-analogue of the class of subset incidences. -
7 Combinatorial Geometry
7 Combinatorial Geometry Planes Incidence Structure: An incidence structure is a triple (V; B; ∼) so that V; B are disjoint sets and ∼ is a relation on V × B. We call elements of V points, elements of B blocks or lines and we associate each line with the set of points incident with it. So, if p 2 V and b 2 B satisfy p ∼ b we say that p is contained in b and write p 2 b and if b; b0 2 B we let b \ b0 = fp 2 P : p ∼ b; p ∼ b0g. Levi Graph: If (V; B; ∼) is an incidence structure, the associated Levi Graph is the bipartite graph with bipartition (V; B) and incidence given by ∼. Parallel: We say that the lines b; b0 are parallel, and write bjjb0 if b \ b0 = ;. Affine Plane: An affine plane is an incidence structure P = (V; B; ∼) which satisfies the following properties: (i) Any two distinct points are contained in exactly one line. (ii) If p 2 V and ` 2 B satisfy p 62 `, there is a unique line containing p and parallel to `. (iii) There exist three points not all contained in a common line. n Line: Let ~u;~v 2 F with ~v 6= 0 and let L~u;~v = f~u + t~v : t 2 Fg. Any set of the form L~u;~v is called a line in Fn. AG(2; F): We define AG(2; F) to be the incidence structure (V; B; ∼) where V = F2, B is the set of lines in F2 and if v 2 V and ` 2 B we define v ∼ ` if v 2 `. -
Contents 5 Eigenvalues and Diagonalization
Linear Algebra (part 5): Eigenvalues and Diagonalization (by Evan Dummit, 2017, v. 1.50) Contents 5 Eigenvalues and Diagonalization 1 5.1 Eigenvalues, Eigenvectors, and The Characteristic Polynomial . 1 5.1.1 Eigenvalues and Eigenvectors . 2 5.1.2 Eigenvalues and Eigenvectors of Matrices . 3 5.1.3 Eigenspaces . 6 5.2 Diagonalization . 9 5.3 Applications of Diagonalization . 14 5.3.1 Transition Matrices and Incidence Matrices . 14 5.3.2 Systems of Linear Dierential Equations . 16 5.3.3 Non-Diagonalizable Matrices and the Jordan Canonical Form . 19 5 Eigenvalues and Diagonalization In this chapter, we will discuss eigenvalues and eigenvectors: these are characteristic values (and characteristic vectors) associated to a linear operator T : V ! V that will allow us to study T in a particularly convenient way. Our ultimate goal is to describe methods for nding a basis for V such that the associated matrix for T has an especially simple form. We will rst describe diagonalization, the procedure for (trying to) nd a basis such that the associated matrix for T is a diagonal matrix, and characterize the linear operators that are diagonalizable. Then we will discuss a few applications of diagonalization, including the Cayley-Hamilton theorem that any matrix satises its characteristic polynomial, and close with a brief discussion of non-diagonalizable matrices. 5.1 Eigenvalues, Eigenvectors, and The Characteristic Polynomial • Suppose that we have a linear transformation T : V ! V from a (nite-dimensional) vector space V to itself. We would like to determine whether there exists a basis of such that the associated matrix β is a β V [T ]β diagonal matrix. -
Eigenvalues of Graphs
Eigenvalues of graphs L¶aszl¶oLov¶asz November 2007 Contents 1 Background from linear algebra 1 1.1 Basic facts about eigenvalues ............................. 1 1.2 Semide¯nite matrices .................................. 2 1.3 Cross product ...................................... 4 2 Eigenvalues of graphs 5 2.1 Matrices associated with graphs ............................ 5 2.2 The largest eigenvalue ................................. 6 2.2.1 Adjacency matrix ............................... 6 2.2.2 Laplacian .................................... 7 2.2.3 Transition matrix ................................ 7 2.3 The smallest eigenvalue ................................ 7 2.4 The eigenvalue gap ................................... 9 2.4.1 Expanders .................................... 10 2.4.2 Edge expansion (conductance) ........................ 10 2.4.3 Random walks ................................. 14 2.5 The number of di®erent eigenvalues .......................... 16 2.6 Spectra of graphs and optimization .......................... 18 Bibliography 19 1 Background from linear algebra 1.1 Basic facts about eigenvalues Let A be an n £ n real matrix. An eigenvector of A is a vector such that Ax is parallel to x; in other words, Ax = ¸x for some real or complex number ¸. This number ¸ is called the eigenvalue of A belonging to eigenvector v. Clearly ¸ is an eigenvalue i® the matrix A ¡ ¸I is singular, equivalently, i® det(A ¡ ¸I) = 0. This is an algebraic equation of degree n for ¸, and hence has n roots (with multiplicity). The trace of the square matrix A = (Aij ) is de¯ned as Xn tr(A) = Aii: i=1 1 The trace of A is the sum of the eigenvalues of A, each taken with the same multiplicity as it occurs among the roots of the equation det(A ¡ ¸I) = 0. If the matrix A is symmetric, then its eigenvalues and eigenvectors are particularly well behaved. -
Notes on the Proof of the Van Der Waerden Permanent Conjecture
Iowa State University Capstones, Theses and Creative Components Dissertations Spring 2018 Notes on the proof of the van der Waerden permanent conjecture Vicente Valle Martinez Iowa State University Follow this and additional works at: https://lib.dr.iastate.edu/creativecomponents Part of the Discrete Mathematics and Combinatorics Commons Recommended Citation Valle Martinez, Vicente, "Notes on the proof of the van der Waerden permanent conjecture" (2018). Creative Components. 12. https://lib.dr.iastate.edu/creativecomponents/12 This Creative Component is brought to you for free and open access by the Iowa State University Capstones, Theses and Dissertations at Iowa State University Digital Repository. It has been accepted for inclusion in Creative Components by an authorized administrator of Iowa State University Digital Repository. For more information, please contact [email protected]. Notes on the proof of the van der Waerden permanent conjecture by Vicente Valle Martinez A Creative Component submitted to the graduate faculty in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE Major: Mathematics Program of Study Committee: Sung Yell Song, Major Professor Steve Butler Jonas Hartwig Leslie Hogben Iowa State University Ames, Iowa 2018 Copyright c Vicente Valle Martinez, 2018. All rights reserved. ii TABLE OF CONTENTS LIST OF FIGURES . iii ACKNOWLEDGEMENTS . iv ABSTRACT . .v CHAPTER 1. Introduction and Preliminaries . .1 1.1 Combinatorial interpretations . .1 1.2 Computational properties of permanents . .4 1.3 Computational complexity in computing permanents . .8 1.4 The organization of the rest of this component . .9 CHAPTER 2. Applications: Permanents of Special Types of Matrices . 10 2.1 Zeros-and-ones matrices .