Solving Eigenvalues Applications from the Point of Theory of Graphs


 Simon Copeland
 1 years ago
 Views:
Transcription
1 Solving Eigenvalues Applications from the Point of Theory of Graphs Soňa PAVLÍKOVÁ Faculty of Mechatronics, AD University of Trenčín Abstract Eigenvalues of linear mappings are applied in many applications in physics, electrical engineering and in other areas of science, especially in fields of military technologies. The dimer problem will be described in the paper and possible ways to resolve this problem in terms of graphs spectrums will be presented. There are many methods for finding eigenvalues of matrixes, either by decomposition of a matrix to canonical forms or numerical methods. The solution of this problem will be explained from the viewpoint of perspective graph theory. Each matrix can be represented as a labeled graph and eigenvalues of the matrix then correspond to the graph spectrum. In case, where a linear mapping matrix is similar to an unsigned integer matrix, other possibilities of further examination of eigenvalues in terms of graph theory are available. Also the necessary and sufficient condition, when a square matrix is diagonally similar to unsigned matrix, will be introduced. Since the possible range of technical problems coverage is much more over the size of this paper, the concrete application for technical problems will be introduced in the next paper of the author. Key words: dimer problem, graph, spectrum of graphs, eigenvalues of matrices Introduction The solution of technical problems is nowadays as close to orthodox mathematics as have never been before. For example, a lot of designers and scientists are using finite elements or other numerical methods with only basic (if any) knowledge of the mathematical background of numerical mathematics and partial differential equations. The previously mentioned example covers the most popular and widespread conjunction of mathematics to designers everydays life. Nevertheless, there are other (sometimes strong, sometimes week) influences of mathematics one of a few explored area is the graph theory. Before the graph theory began wellknown, the technical problems have been rewritten according to the given (and usually very complicated) rules into the form of matrixes. One of the most important indicators is the eigenvalues of the matrixes. The consecutive analysis is always based on theirs evaluation. Any technical problem (could be based on electrotechnical engineering, mechanics, micro and nano tech, civil engineering or chemistry) can be more or less easily transformed into the structure of oriented graphs. These can be used either for storing the data in less space and time consuming way; or, and this is the way of this paper) for analysis and testing the structure properties. When transforming the technical problem into the form of oriented
2 graphs, the eigenvalues of matrixes can be obtained in the simpler way and this will spare designer s time and performance of CPU. The important technique used in this kind of analysis is called linear mapping. Eigenvalues and eigenvectors of linear mappings are widely applied in solving of many applications in electrical engineering, chemistry or economy. Linear mappings can be characterized as a mapping between two vector spaces that retain the coefficients of linear combination of vectors in two given spaces. Every linear mapping has its own matrix representation. There are many ways how to solve the problem of eigenvalues. Methods of linear algebra use the similarity of matrixes. It is well known that similar matrixes have the same eigenvalues. Every square matrix is similar to the matrix in rational canonical form. Eigenvalues of a matrix can be solved by numerical methods. The eingenvalue of A are the roots of the characteristic polynomial of A. Once the characteristic polynomial is known, the roots can be solved by using subroutines for solving the roots of polynomials. However a polynomial may be ill  conditioned in the sense that the small changes in coefficients may cause large changes in one or more roots. Hence this procedure of computing the eigenvalues may change a well  conditioned problem into an ill conditioned problem and should be avoided. Therefore, it is appropriate to look for other ways to solve this problem. The problem of eigenvalues can be also solved through the graph theory, which brings up many nontraditional solutions. Background of the Problem 1,..., n Let G is a finite nonoriented graph without loops with a vertex set V G v v and an edge set of graph G is E G (multiple edges are allowed). Adjacency matrix A a ij a square matrix of n x n, whose element vertex v i with the vertex a ij is equal to the number of edges connecting the v j. Obviously, the graph G is explicitly determined (but isomorphism) by matrix A, but vice versa it is not true  matrix A depends on the labeling of the vertixes. Therefore, we can assign the matrix class M=M(A ) to the graph G, where two matrixes A and B belong in this class if and only if, where there is such a permutation matrix 1 P that B P AP. Important invariant of the class M is the characteristic polynomial P I A, where i, i 1,2,..., n is the root of the equation PG ( ) 0 and n. For analyzing eigenvalues of a matrix using assigned graph, it is important that the matrix is similar to nonnegative matrix. In the results we will present the condition, when the matrix is similar to nonnegative matrix. Bipartite graphs play an important role in the following motivation example. Graph G will be called bipartite if it does not contain a circle of an odd length. Vertex set of bipartite graph can be G det( ), respectively Sp( G) 1, 2,..., n divided into two disjoint subsets R and C, that R C V( G) and no two vertixes from the set R, respectively C are connected by an edge. F factor of graph G is a such subgraph of graph G, that V (F) = V (G) and E( F) E( G). The Dimer Problem The spectra of graphs, or the spectra of certain matrices which are closely related to the adjacency matrices appear in a number of problems in statistical physics [1], [2], [3]. We shall describe so called dimer problem.
3 The dimer problem is related to the investigation of the thermodynamic properties of a system of diatomic molecules ( dimers ) adsorbed on the surface of a crystal. The most favorable points for the absorption of atoms on such surface form a twodimensional lattice, and a dimer can occupy two neighboring points. It is necessary to count all ways in which dimers can be arranged on the lattice without overlapping each other, so that every lattice point is occupied. The dimer problem on a square lattice is equivalent to the problem of enumerating all 1 2 ways in which a chessboard of dimension n x n (n being even) can be covered by 2 n dominoes, so that each domino covers two adjacent squares of the chessboard and that all squares are so covered. A graph can be associated with a given adsorption surface. The vertices of the graph represent the points which are the most favorable for adsorption. Two vertices are adjacent if and only if the corresponding points can be occupied by a dimer. In this manner an arrangement of dimers on the surface determines a factor in the corresponding graph, and vice versa. Thus, the dimer problem is reduced to the task of determining the number of factors in a graph. Such assigned graphs are bipartite and fall into special classes of graphs, which we mark A. The reader will find a detailed description of the class A of graphs for example in [4]. The basic Theorem [4] can be stated as:. Let G A.. The number of factors of G is equal to the product of all non negative eigenvalues of G. Results and Summary Theorem 1 can be stated as: Let A is a square matrix of order n. Then A is diagonally similar to unsigned matrix if and only if there is a decomposition of the index set 1,2,...,n for nonempty sets of and with the property that A and A A and A are nonpositive matrixes. Lema 1: are nonnegative matrixes and Proof of the Theorem 1 is a consequence from the following statements: Let A a ij and B b ij same zero points, i.e. aij 0 if and only if bij 0. Lema 2: every i, j: Let A a ij and B b ij are diagonally similar matrixes. Then A and B have the are square matrixes, with the following properties for
4 (i) if bij 0 then aij 0 (ii) if aij c, where c is a real number different from zero, then bij aij. Then, diagonal similarity of matrix B with unsigned matrix implies a diagonal similarity of matrix A with nonnegative matrix. Lema 3: Let A be a square matrix of order n. Then A is diagonally similar with some unsigned matrix if and only if A is (0, 1, 1)  diagonally similar with some nonnegative matrix. The following concequence can be derived from Theorem 1. Let A be a square matrix of order n. Then A is diagonally similar with some nonnegative matrix if and only if there is such permutation matrix P that AA PAP AA 21 2 where A 11 and A 22 are nonnegative square matrixes and matrixes A 12 and A 21 are nonpositive. Conclusions The scope of this paper shows options available when using the graph spectrums. Spectra of graphs have appeared frequently in mathematical literature since the fundamental papers [5]. Even earlier, starting from the thesis of [6] in 1931, theoretical chemists were interested in graph spectra, although they used different terminology. The possibilities of solving eigenvalues using graph theory have been outlined in the paper. Problems of graphs spectra open up many so far unexplored possibilities in applications and give an unusual insight into many problems of physics, chemistry, electrical engineering etc. very commonly used to solve the problems of special and military technologies. The aim of this paper is to give the reader the necessary mathematical background of the presented objectives. The concrete technical applications will be introduced in the next paper. References 1. KASTELEYN, P. W. Graph Theory and theoretical Physics. London  New York : Academic Press, Graph theory and crystal physics, pp MONTROL, E. W. Applied Combinatorial Mathematics. London  New York  Sydney: Interscience, John Wiley &Sons, Latice statistics, pp PERCUS, J. K. Combinatorial Methods. Berlin  Heidelberg  New York : Springer  Verlag, CVETKOVIĆ, D. M.; DOOB, M.; SACHS, H. Spectra of Graphs : Theory and Application. Second edition. Berlin: VEB Deutscher Verlag der Wissenschaften, p. 5. SINOGOWITZ, U., COLLATZ, L. Spektren endlicher Grafen. Abh. Math. Sem. Univ. Hamburg. 1957, 21, pp HÜCKEL, E. Quantentheoretische Beiträge zum Benzolproblem. Z. Phys , 70, pp
5 7. MATEJDES, M. Aplikovaná matematika. Zvolen: MAT  CENTRUM, p. ISBN
COMBINATORIAL CHESSBOARD REARRANGEMENTS
COMBINATORIAL CHESSBOARD REARRANGEMENTS DARYL DEFORD Abstract. Many problems concerning tilings of rectangular boards are of significant combinatorial interest. In this paper we introduce a similar type
More informationA CHARACTERIZATION OF TREE TYPE
A CHARACTERIZATION OF TREE TYPE LON H MITCHELL Abstract Let L(G) be the Laplacian matrix of a simple graph G The characteristic valuation associated with the algebraic connectivity a(g) is used in classifying
More informationby the matrix A results in a vector which is a reflection of the given
Eigenvalues & Eigenvectors Example Suppose Then So, geometrically, multiplying a vector in by the matrix A results in a vector which is a reflection of the given vector about the yaxis We observe that
More informationChapter 6. Orthogonality
6.3 Orthogonal Matrices 1 Chapter 6. Orthogonality 6.3 Orthogonal Matrices Definition 6.4. An n n matrix A is orthogonal if A T A = I. Note. We will see that the columns of an orthogonal matrix must be
More informationSimilarity and Diagonalization. Similar Matrices
MATH022 Linear Algebra Brief lecture notes 48 Similarity and Diagonalization Similar Matrices Let A and B be n n matrices. We say that A is similar to B if there is an invertible n n matrix P such that
More informationMATH 423 Linear Algebra II Lecture 38: Generalized eigenvectors. Jordan canonical form (continued).
MATH 423 Linear Algebra II Lecture 38: Generalized eigenvectors Jordan canonical form (continued) Jordan canonical form A Jordan block is a square matrix of the form λ 1 0 0 0 0 λ 1 0 0 0 0 λ 0 0 J = 0
More informationThe Open University s repository of research publications and other research outputs
Open Research Online The Open University s repository of research publications and other research outputs The degreediameter problem for circulant graphs of degree 8 and 9 Journal Article How to cite:
More information10. Graph Matrices Incidence Matrix
10 Graph Matrices Since a graph is completely determined by specifying either its adjacency structure or its incidence structure, these specifications provide far more efficient ways of representing a
More informationKragujevac Trees and Their Energy
SCIENTIFIC PUBLICATIONS OF THE STATE UNIVERSITY OF NOVI PAZAR SER. A: APPL. MATH. INFORM. AND MECH. vol. 6, 2 (2014), 7179. Kragujevac Trees and Their Energy Ivan Gutman INVITED PAPER Abstract: The Kragujevac
More informationClassification of Cartan matrices
Chapter 7 Classification of Cartan matrices In this chapter we describe a classification of generalised Cartan matrices This classification can be compared as the rough classification of varieties in terms
More informationSHARP BOUNDS FOR THE SUM OF THE SQUARES OF THE DEGREES OF A GRAPH
31 Kragujevac J. Math. 25 (2003) 31 49. SHARP BOUNDS FOR THE SUM OF THE SQUARES OF THE DEGREES OF A GRAPH Kinkar Ch. Das Department of Mathematics, Indian Institute of Technology, Kharagpur 721302, W.B.,
More informationMarkov Chains, Stochastic Processes, and Advanced Matrix Decomposition
Markov Chains, Stochastic Processes, and Advanced Matrix Decomposition Jack Gilbert Copyright (c) 2014 Jack Gilbert. Permission is granted to copy, distribute and/or modify this document under the terms
More informationCOMBINATORIAL PROPERTIES OF THE HIGMANSIMS GRAPH. 1. Introduction
COMBINATORIAL PROPERTIES OF THE HIGMANSIMS GRAPH ZACHARY ABEL 1. Introduction In this survey we discuss properties of the HigmanSims graph, which has 100 vertices, 1100 edges, and is 22 regular. In fact
More informationUSE OF EIGENVALUES AND EIGENVECTORS TO ANALYZE BIPARTIVITY OF NETWORK GRAPHS
USE OF EIGENVALUES AND EIGENVECTORS TO ANALYZE BIPARTIVITY OF NETWORK GRAPHS Natarajan Meghanathan Jackson State University, 1400 Lynch St, Jackson, MS, USA natarajan.meghanathan@jsums.edu ABSTRACT This
More informationAN UPPER BOUND ON THE LAPLACIAN SPECTRAL RADIUS OF THE SIGNED GRAPHS
Discussiones Mathematicae Graph Theory 28 (2008 ) 345 359 AN UPPER BOUND ON THE LAPLACIAN SPECTRAL RADIUS OF THE SIGNED GRAPHS HongHai Li College of Mathematic and Information Science Jiangxi Normal University
More informationZachary Monaco Georgia College Olympic Coloring: Go For The Gold
Zachary Monaco Georgia College Olympic Coloring: Go For The Gold Coloring the vertices or edges of a graph leads to a variety of interesting applications in graph theory These applications include various
More informationTopological Entropy of Golden Mean Lookalike Shift Spaces
International Journal of Mathematics and Statistics Invention (IJMSI) EISSN: 2321 4767 PISSN: 23214759 Volume 2 Issue 7 July 2014 PP0510 Topological Entropy of Golden Mean Lookalike Shift Spaces Khundrakpam
More informationDATA ANALYSIS II. Matrix Algorithms
DATA ANALYSIS II Matrix Algorithms Similarity Matrix Given a dataset D = {x i }, i=1,..,n consisting of n points in R d, let A denote the n n symmetric similarity matrix between the points, given as where
More informationUNIT 2 MATRICES  I 2.0 INTRODUCTION. Structure
UNIT 2 MATRICES  I Matrices  I Structure 2.0 Introduction 2.1 Objectives 2.2 Matrices 2.3 Operation on Matrices 2.4 Invertible Matrices 2.5 Systems of Linear Equations 2.6 Answers to Check Your Progress
More informationMath 4707: Introduction to Combinatorics and Graph Theory
Math 4707: Introduction to Combinatorics and Graph Theory Lecture Addendum, November 3rd and 8th, 200 Counting Closed Walks and Spanning Trees in Graphs via Linear Algebra and Matrices Adjacency Matrices
More informationSplit Nonthreshold Laplacian Integral Graphs
Split Nonthreshold Laplacian Integral Graphs Stephen Kirkland University of Regina, Canada kirkland@math.uregina.ca Maria Aguieiras Alvarez de Freitas Federal University of Rio de Janeiro, Brazil maguieiras@im.ufrj.br
More informationContinuity of the Perron Root
Linear and Multilinear Algebra http://dx.doi.org/10.1080/03081087.2014.934233 ArXiv: 1407.7564 (http://arxiv.org/abs/1407.7564) Continuity of the Perron Root Carl D. Meyer Department of Mathematics, North
More informationMTH304: Honors Algebra II
MTH304: Honors Algebra II This course builds upon algebraic concepts covered in Algebra. Students extend their knowledge and understanding by solving openended problems and thinking critically. Topics
More informationZero: If P is a polynomial and if c is a number such that P (c) = 0 then c is a zero of P.
MATH 11011 FINDING REAL ZEROS KSU OF A POLYNOMIAL Definitions: Polynomial: is a function of the form P (x) = a n x n + a n 1 x n 1 + + a x + a 1 x + a 0. The numbers a n, a n 1,..., a 1, a 0 are called
More informationMathematics Course 111: Algebra I Part IV: Vector Spaces
Mathematics Course 111: Algebra I Part IV: Vector Spaces D. R. Wilkins Academic Year 19967 9 Vector Spaces A vector space over some field K is an algebraic structure consisting of a set V on which are
More informationLinear Algebra and TI 89
Linear Algebra and TI 89 Abdul Hassen and Jay Schiffman This short manual is a quick guide to the use of TI89 for Linear Algebra. We do this in two sections. In the first section, we will go over the editing
More informationDO NOT REDISTRIBUTE THIS SOLUTION FILE
Professor Kindred Math 04 Graph Theory Homework 7 Solutions April 3, 03 Introduction to Graph Theory, West Section 5. 0, variation of 5, 39 Section 5. 9 Section 5.3 3, 8, 3 Section 7. Problems you should
More informationPart 2: Community Detection
Chapter 8: Graph Data Part 2: Community Detection Based on Leskovec, Rajaraman, Ullman 2014: Mining of Massive Datasets Big Data Management and Analytics Outline Community Detection  Social networks 
More informationIRREDUCIBLE OPERATOR SEMIGROUPS SUCH THAT AB AND BA ARE PROPORTIONAL. 1. Introduction
IRREDUCIBLE OPERATOR SEMIGROUPS SUCH THAT AB AND BA ARE PROPORTIONAL R. DRNOVŠEK, T. KOŠIR Dedicated to Prof. Heydar Radjavi on the occasion of his seventieth birthday. Abstract. Let S be an irreducible
More informationTricyclic biregular graphs whose energy exceeds the number of vertices
MATHEMATICAL COMMUNICATIONS 213 Math. Commun., Vol. 15, No. 1, pp. 213222 (2010) Tricyclic biregular graphs whose energy exceeds the number of vertices Snježana Majstorović 1,, Ivan Gutman 2 and Antoaneta
More informationUniversity of Lille I PC first year list of exercises n 7. Review
University of Lille I PC first year list of exercises n 7 Review Exercise Solve the following systems in 4 different ways (by substitution, by the Gauss method, by inverting the matrix of coefficients
More informationSign pattern matrices that admit M, N, P or inverse M matrices
Sign pattern matrices that admit M, N, P or inverse M matrices C Mendes Araújo, Juan R Torregrosa CMAT  Centro de Matemática / Dpto de Matemática Aplicada Universidade do Minho / Universidad Politécnica
More information13 Solutions for Section 6
13 Solutions for Section 6 Exercise 6.2 Draw up the group table for S 3. List, giving each as a product of disjoint cycles, all the permutations in S 4. Determine the order of each element of S 4. Solution
More informationLinear Algebra and its Applications
Linear Algebra and its Applications 438 2013) 1393 1397 Contents lists available at SciVerse ScienceDirect Linear Algebra and its Applications journal homepage: www.elsevier.com/locate/laa Note on the
More informationPolytope Examples (PolyComp Fukuda) Matching Polytope 1
Polytope Examples (PolyComp Fukuda) Matching Polytope 1 Matching Polytope Let G = (V,E) be a graph. A matching in G is a subset of edges M E such that every vertex meets at most one member of M. A matching
More informationNotes on Determinant
ENGG2012B Advanced Engineering Mathematics Notes on Determinant Lecturer: Kenneth Shum Lecture 918/02/2013 The determinant of a system of linear equations determines whether the solution is unique, without
More informationCHAPTER 12 MOLECULAR SYMMETRY
CHAPTER 12 MOLECULAR SYMMETRY In many cases, the symmetry of a molecule provides a great deal of information about its quantum states, even without a detailed solution of the Schrödinger equation. A geometrical
More informationCycles in a Graph Whose Lengths Differ by One or Two
Cycles in a Graph Whose Lengths Differ by One or Two J. A. Bondy 1 and A. Vince 2 1 LABORATOIRE DE MATHÉMATIQUES DISCRÉTES UNIVERSITÉ CLAUDEBERNARD LYON 1 69622 VILLEURBANNE, FRANCE 2 DEPARTMENT OF MATHEMATICS
More informationFacts About Eigenvalues
Facts About Eigenvalues By Dr David Butler Definitions Suppose A is an n n matrix An eigenvalue of A is a number λ such that Av = λv for some nonzero vector v An eigenvector of A is a nonzero vector v
More informationAlgebraic Concepts Algebraic Concepts Writing
Curriculum Guide: Algebra 2/Trig (AR) 2 nd Quarter 8/7/2013 2 nd Quarter, Grade 912 GRADE 912 Unit of Study: Matrices Resources: Textbook: Algebra 2 (Holt, Rinehart & Winston), Ch. 4 Length of Study:
More informationDETERMINANTS IN THE KRONECKER PRODUCT OF MATRICES: THE INCIDENCE MATRIX OF A COMPLETE GRAPH
DETERMINANTS IN THE KRONECKER PRODUCT OF MATRICES: THE INCIDENCE MATRIX OF A COMPLETE GRAPH CHRISTOPHER RH HANUSA AND THOMAS ZASLAVSKY Abstract We investigate the least common multiple of all subdeterminants,
More informationYILUN SHANG. e λi. i=1
LOWER BOUNDS FOR THE ESTRADA INDEX OF GRAPHS YILUN SHANG Abstract. Let G be a graph with n vertices and λ 1,λ,...,λ n be its eigenvalues. The Estrada index of G is defined as EE(G = n eλ i. In this paper,
More informationText: A Graphical Approach to College Algebra (Hornsby, Lial, Rockswold)
Students will take Self Tests covering the topics found in Chapter R (Reference: Basic Algebraic Concepts) and Chapter 1 (Linear Functions, Equations, and Inequalities). If any deficiencies are revealed,
More informationMATH 240 Fall, Chapter 1: Linear Equations and Matrices
MATH 240 Fall, 2007 Chapter Summaries for Kolman / Hill, Elementary Linear Algebra, 9th Ed. written by Prof. J. Beachy Sections 1.1 1.5, 2.1 2.3, 4.2 4.9, 3.1 3.5, 5.3 5.5, 6.1 6.3, 6.5, 7.1 7.3 DEFINITIONS
More informationSolution based on matrix technique Rewrite. ) = 8x 2 1 4x 1x 2 + 5x x1 2x 2 2x 1 + 5x 2
8.2 Quadratic Forms Example 1 Consider the function q(x 1, x 2 ) = 8x 2 1 4x 1x 2 + 5x 2 2 Determine whether q(0, 0) is the global minimum. Solution based on matrix technique Rewrite q( x1 x 2 = x1 ) =
More information7  Linear Transformations
7  Linear Transformations Mathematics has as its objects of study sets with various structures. These sets include sets of numbers (such as the integers, rationals, reals, and complexes) whose structure
More informationOrthogonal Diagonalization of Symmetric Matrices
MATH10212 Linear Algebra Brief lecture notes 57 Gram Schmidt Process enables us to find an orthogonal basis of a subspace. Let u 1,..., u k be a basis of a subspace V of R n. We begin the process of finding
More informationMath 115A HW4 Solutions University of California, Los Angeles. 5 2i 6 + 4i. (5 2i)7i (6 + 4i)( 3 + i) = 35i + 14 ( 22 6i) = 36 + 41i.
Math 5A HW4 Solutions September 5, 202 University of California, Los Angeles Problem 4..3b Calculate the determinant, 5 2i 6 + 4i 3 + i 7i Solution: The textbook s instructions give us, (5 2i)7i (6 + 4i)(
More informationMathematics of Cryptography
CHAPTER 2 Mathematics of Cryptography Part I: Modular Arithmetic, Congruence, and Matrices Objectives This chapter is intended to prepare the reader for the next few chapters in cryptography. The chapter
More informationWe call this set an ndimensional parallelogram (with one vertex 0). We also refer to the vectors x 1,..., x n as the edges of P.
Volumes of parallelograms 1 Chapter 8 Volumes of parallelograms In the present short chapter we are going to discuss the elementary geometrical objects which we call parallelograms. These are going to
More informationBlock designs/1. 1 Background
Block designs 1 Background In a typical experiment, we have a set Ω of experimental units or plots, and (after some preparation) we make a measurement on each plot (for example, the yield of the plot).
More informationReciprocal Space and Brillouin Zones in Two and Three Dimensions As briefly stated at the end of the first section, Bloch s theorem has the following
Reciprocal Space and Brillouin Zones in Two and Three Dimensions As briefly stated at the end of the first section, Bloch s theorem has the following form in two and three dimensions: k (r + R) = e 2 ik
More informationLecture notes from Foundations of Markov chain Monte Carlo methods University of Chicago, Spring 2002 Lecture 1, March 29, 2002
Lecture notes from Foundations of Markov chain Monte Carlo methods University of Chicago, Spring 2002 Lecture 1, March 29, 2002 Eric Vigoda Scribe: Varsha Dani & Tom Hayes 1.1 Introduction The aim of this
More informationSummary of week 8 (Lectures 22, 23 and 24)
WEEK 8 Summary of week 8 (Lectures 22, 23 and 24) This week we completed our discussion of Chapter 5 of [VST] Recall that if V and W are inner product spaces then a linear map T : V W is called an isometry
More informationThe Puzzle Layout Problem
The Puzzle Layout Problem Kozo Sugiyama 1, SeokHee Hong 2, and Atsuhiko Maeda 3 1 School of Knowledge Science, Japan Advanced Institute of Science and Technology, Asahidai 11, Tatsunokuchi, Nomi, Ishikawa,
More information4. Factor polynomials over complex numbers, describe geometrically, and apply to realworld situations. 5. Determine and apply relationships among syn
I The Real and Complex Number Systems 1. Identify subsets of complex numbers, and compare their structural characteristics. 2. Compare and contrast the properties of real numbers with the properties of
More informationResearch experiences for undergraduate faculty
Research experiences for undergraduate faculty organized by Leslie Hogben and Roselyn Williams Workshop Summary On July 2024, 2009, the American Institute of Mathematics (AIM), with support from the National
More informationMathematics (MAT) MAT 061 Basic Euclidean Geometry 3 Hours. MAT 051 PreAlgebra 4 Hours
MAT 051 PreAlgebra Mathematics (MAT) MAT 051 is designed as a review of the basic operations of arithmetic and an introduction to algebra. The student must earn a grade of C or in order to enroll in MAT
More informationApproximating the entropy of a 2dimensional shift of finite type
Approximating the entropy of a dimensional shift of finite type Tirasan Khandhawit c 4 July 006 Abstract. In this paper, we extend the method used to compute entropy of dimensional subshift and the technique
More informationBindel, Fall 2012 Matrix Computations (CS 6210) Week 8: Friday, Oct 12
Why eigenvalues? Week 8: Friday, Oct 12 I spend a lot of time thinking about eigenvalue problems. In part, this is because I look for problems that can be solved via eigenvalues. But I might have fewer
More informationMAT2400 Analysis I. A brief introduction to proofs, sets, and functions
MAT2400 Analysis I A brief introduction to proofs, sets, and functions In Analysis I there is a lot of manipulations with sets and functions. It is probably also the first course where you have to take
More informationLecture 1 (Review of High School Math: Functions and Models) Introduction: Numbers and their properties
Lecture 1 (Review of High School Math: Functions and Models) Introduction: Numbers and their properties Addition: (1) (Associative law) If a, b, and c are any numbers, then ( ) ( ) (2) (Existence of an
More informationInner Product Spaces and Orthogonality
Inner Product Spaces and Orthogonality week 34 Fall 2006 Dot product of R n The inner product or dot product of R n is a function, defined by u, v a b + a 2 b 2 + + a n b n for u a, a 2,, a n T, v b,
More information2.1: Determinants by Cofactor Expansion. Math 214 Chapter 2 Notes and Homework. Evaluate a Determinant by Expanding by Cofactors
2.1: Determinants by Cofactor Expansion Math 214 Chapter 2 Notes and Homework Determinants The minor M ij of the entry a ij is the determinant of the submatrix obtained from deleting the i th row and the
More informationChapter 15 Introduction to Linear Programming
Chapter 15 Introduction to Linear Programming An Introduction to Optimization Spring, 2014 WeiTa Chu 1 Brief History of Linear Programming The goal of linear programming is to determine the values of
More informationThe Characteristic Polynomial
Physics 116A Winter 2011 The Characteristic Polynomial 1 Coefficients of the characteristic polynomial Consider the eigenvalue problem for an n n matrix A, A v = λ v, v 0 (1) The solution to this problem
More informationWeek 910: Recurrence Relations and Generating Functions
Week 910: Recurrence Relations and Generating Functions March 31, 2015 1 Some number sequences An infinite sequence (or just a sequence for short is an ordered array a 0, a 1, a 2,..., a n,... of countably
More informationSukGeun Hwang and JinWoo Park
Bull. Korean Math. Soc. 43 (2006), No. 3, pp. 471 478 A NOTE ON PARTIAL SIGNSOLVABILITY SukGeun Hwang and JinWoo Park Abstract. In this paper we prove that if Ax = b is a partial signsolvable linear
More informationAlgebra 1 Course Title
Algebra 1 Course Title Course wide 1. What patterns and methods are being used? Course wide 1. Students will be adept at solving and graphing linear and quadratic equations 2. Students will be adept
More informationLesson 3. Algebraic graph theory. Sergio Barbarossa. Rome  February 2010
Lesson 3 Algebraic graph theory Sergio Barbarossa Basic notions Definition: A directed graph (or digraph) composed by a set of vertices and a set of edges We adopt the convention that the information flows
More informationMath 1050 Khan Academy Extra Credit Algebra Assignment
Math 1050 Khan Academy Extra Credit Algebra Assignment KhanAcademy.org offers over 2,700 instructional videos, including hundreds of videos teaching algebra concepts, and corresponding problem sets. In
More informationSTUDY GUIDE LINEAR ALGEBRA. David C. Lay University of Maryland College Park AND ITS APPLICATIONS THIRD EDITION UPDATE
STUDY GUIDE LINEAR ALGEBRA AND ITS APPLICATIONS THIRD EDITION UPDATE David C. Lay University of Maryland College Park Copyright 2006 Pearson AddisonWesley. All rights reserved. Reproduced by Pearson AddisonWesley
More informationON THE COEFFICIENTS OF THE LINKING POLYNOMIAL
ADSS, Volume 3, Number 3, 2013, Pages 4556 2013 Aditi International ON THE COEFFICIENTS OF THE LINKING POLYNOMIAL KOKO KALAMBAY KAYIBI Abstract Let i j T( M; = tijx y be the Tutte polynomial of the matroid
More informationLinear Programming I
Linear Programming I November 30, 2003 1 Introduction In the VCR/guns/nuclear bombs/napkins/star wars/professors/butter/mice problem, the benevolent dictator, Bigus Piguinus, of south Antarctica penguins
More informationNETZCOPE  a tool to analyze and display complex R&D collaboration networks
The Task Concepts from Spectral Graph Theory EU R&D Network Analysis Netzcope Screenshots NETZCOPE  a tool to analyze and display complex R&D collaboration networks L. Streit & O. Strogan BiBoS, Univ.
More informationQuadratic Equations in Finite Fields of Characteristic 2
Quadratic Equations in Finite Fields of Characteristic 2 Klaus Pommerening May 2000 english version February 2012 Quadratic equations over fields of characteristic 2 are solved by the well known quadratic
More information3.1 State Space Models
31 State Space Models In this section we study state space models of continuoustime linear systems The corresponding results for discretetime systems, obtained via duality with the continuoustime models,
More informationOverview of Math Standards
Algebra 2 Welcome to math curriculum design maps for Manhattan Ogden USD 383, striving to produce learners who are: Effective Communicators who clearly express ideas and effectively communicate with diverse
More informationMath 550 Notes. Chapter 7. Jesse Crawford. Department of Mathematics Tarleton State University. Fall 2010
Math 550 Notes Chapter 7 Jesse Crawford Department of Mathematics Tarleton State University Fall 2010 (Tarleton State University) Math 550 Chapter 7 Fall 2010 1 / 34 Outline 1 SelfAdjoint and Normal Operators
More information15.062 Data Mining: Algorithms and Applications Matrix Math Review
.6 Data Mining: Algorithms and Applications Matrix Math Review The purpose of this document is to give a brief review of selected linear algebra concepts that will be useful for the course and to develop
More informationNonlinear Iterative Partial Least Squares Method
Numerical Methods for Determining Principal Component Analysis Abstract Factors Béchu, S., RichardPlouet, M., Fernandez, V., Walton, J., and Fairley, N. (2016) Developments in numerical treatments for
More informationSocial Media Mining. Network Measures
Klout Measures and Metrics 22 Why Do We Need Measures? Who are the central figures (influential individuals) in the network? What interaction patterns are common in friends? Who are the likeminded users
More informationNotes on Orthogonal and Symmetric Matrices MENU, Winter 2013
Notes on Orthogonal and Symmetric Matrices MENU, Winter 201 These notes summarize the main properties and uses of orthogonal and symmetric matrices. We covered quite a bit of material regarding these topics,
More informationTransportation Polytopes: a Twenty year Update
Transportation Polytopes: a Twenty year Update Jesús Antonio De Loera University of California, Davis Based on various papers joint with R. Hemmecke, E.Kim, F. Liu, U. Rothblum, F. Santos, S. Onn, R. Yoshida,
More informationThe determinant of a skewsymmetric matrix is a square. This can be seen in small cases by direct calculation: 0 a. 12 a. a 13 a 24 a 14 a 23 a 14
4 Symplectic groups In this and the next two sections, we begin the study of the groups preserving reflexive sesquilinear forms or quadratic forms. We begin with the symplectic groups, associated with
More informationpp. 4 8: Examples 1 6 Quick Check 1 6 Exercises 1, 2, 20, 42, 43, 64
Semester 1 Text: Chapter 1: Tools of Algebra Lesson 11: Properties of Real Numbers Day 1 Part 1: Graphing and Ordering Real Numbers Part 1: Graphing and Ordering Real Numbers Lesson 12: Algebraic Expressions
More information. 0 1 10 2 100 11 1000 3 20 1 2 3 4 5 6 7 8 9
Introduction The purpose of this note is to find and study a method for determining and counting all the positive integer divisors of a positive integer Let N be a given positive integer We say d is a
More informationDiscrete Mathematics & Mathematical Reasoning Chapter 10: Graphs
Discrete Mathematics & Mathematical Reasoning Chapter 10: Graphs Kousha Etessami U. of Edinburgh, UK Kousha Etessami (U. of Edinburgh, UK) Discrete Mathematics (Chapter 6) 1 / 13 Overview Graphs and Graph
More informationI. BASIC PERRON FROBENIUS THEORY AND INVERSE SPECTRAL PROBLEMS
I. BASIC PERRON FROBENIUS THEORY AND INVERSE SPECTRAL PROBLEMS MIKE BOYLE Contents 1. Introduction 1 2. The primitive case 1 3. Why the Perron Theorem is useful 2 4. A proof of the Perron Theorem 3 5.
More informationMatrix Algebra 2.3 CHARACTERIZATIONS OF INVERTIBLE MATRICES Pearson Education, Inc.
2 Matrix Algebra 2.3 CHARACTERIZATIONS OF INVERTIBLE MATRICES Theorem 8: Let A be a square matrix. Then the following statements are equivalent. That is, for a given A, the statements are either all true
More informationTHE FUNDAMENTAL THEOREM OF ALGEBRA VIA PROPER MAPS
THE FUNDAMENTAL THEOREM OF ALGEBRA VIA PROPER MAPS KEITH CONRAD 1. Introduction The Fundamental Theorem of Algebra says every nonconstant polynomial with complex coefficients can be factored into linear
More informationIntersection Dimension and Maximum Degree
Intersection Dimension and Maximum Degree N.R. Aravind and C.R. Subramanian The Institute of Mathematical Sciences, Taramani, Chennai  600 113, India. email: {nraravind,crs}@imsc.res.in Abstract We show
More informationMATH 304 Linear Algebra Lecture 8: Inverse matrix (continued). Elementary matrices. Transpose of a matrix.
MATH 304 Linear Algebra Lecture 8: Inverse matrix (continued). Elementary matrices. Transpose of a matrix. Inverse matrix Definition. Let A be an n n matrix. The inverse of A is an n n matrix, denoted
More informationIntroduction to Matrix Algebra
Psychology 7291: Multivariate Statistics (Carey) 8/27/98 Matrix Algebra  1 Introduction to Matrix Algebra Definitions: A matrix is a collection of numbers ordered by rows and columns. It is customary
More informationENERGY OF COMPLEMENT OF STARS
J Indones Math Soc Vol 19, No 1 (2013), pp 15 21 ENERGY OF COMPLEMENT OF STARS P R Hampiholi 1, H B Walikar 2, and B S Durgi 3 1 Department of Master of Computer Applications, Gogte Institute of Technology,
More information10.3 POWER METHOD FOR APPROXIMATING EIGENVALUES
55 CHAPTER NUMERICAL METHODS. POWER METHOD FOR APPROXIMATING EIGENVALUES In Chapter 7 we saw that the eigenvalues of an n n matrix A are obtained by solving its characteristic equation n c n n c n n...
More information7 Communication Classes
this version: 26 February 2009 7 Communication Classes Perhaps surprisingly, we can learn much about the longrun behavior of a Markov chain merely from the zero pattern of its transition matrix. In the
More informationNorman Do. Scissors congruence and Hilbert s third problem. What is area?
Norman Do Scissors congruence and Hilbert s third problem What is area? Despite being such a fundamental notion of geometry, the concept of area is very difficult to define. As human beings, we have an
More informationMatrix Norms. Tom Lyche. September 28, Centre of Mathematics for Applications, Department of Informatics, University of Oslo
Matrix Norms Tom Lyche Centre of Mathematics for Applications, Department of Informatics, University of Oslo September 28, 2009 Matrix Norms We consider matrix norms on (C m,n, C). All results holds for
More informationMath 225A, Differential Topology: Homework 3
Math 225A, Differential Topology: Homework 3 Ian Coley October 17, 2013 Problem 1.4.7. Suppose that y is a regular value of f : X Y, where X is compact and dim X = dim Y. Show that f 1 (y) is a finite
More information