Save this PDF as:

Size: px
Start display at page:

Transcription

1 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

2 Happy Birthday, MSG!! Hadamard s theorem and many happy returns... 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 n 1/2, so that det(h) n n/2 ; equality holds if and only if every entry of H is ±1; 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. Remarks HH = ni H 1 = n 1 H H H = ni, so a Hadamard matrix also has orthogonal columns. Changing signs of rows or columns, permuting rows or columns, or transposing preserve the Hadamard property. Examples of Hadamard matrices include ( ) ( ) + + +,, Orders of Hadamard matrices The order of a Hadamard matrix is 1, 2 or a multiple of 4. We can ensure that the first row consists of all +s by column sign changes. Then (assuming at least three rows) we can bring the first three rows into the following shape by column permutations: a b c d Now orthogonality of rows gives a + b = c + d = a + c = b + d = a + d = b + c = n/2, so a = b = c = d = n/4. The Hadamard conjecture Symmetric Hadamard matrices The Hadamard conjecture asserts that a Hadamard matrix exists of every order divisible by 4. The smallest multiple of 4 for which no such matrix is currently known is 668, the value 428 having been settled only in A particularly attractive class of Hadamard matrices are those which are symmetric, have constant diagonal and constant row sum. Such matrices must have square order 4s 2 ; the row sums are ±2s. [For the row sum σ is an eigenvalue of H, and hence σ 2 is an eigenvalue of H 2 = HH : thus σ 2 = n.] They give rise to symmetric 2-(4s 2, 2s 2 ± s, s 2 ± s) designs and strongly regular graphs. In the case where the order is a power of 2, these matrices can be constructed from bent functions (functions on a vector space whose distance from the space of linear functions is maximal). There are connections with coding theory and cryptography.

3 Skew-Hadamard matrices A matrix A is skew if A = A. A Hadamard matrix can t really be skew, since in characteristic not 2, a skew matrix has zero diagonal. So we compromise and define a skew-hadamard matrix H to be one which has constant diagonal +1 and such that H I is skew. The property is preserved by simultaneous row and column sign changes, so we can normalise the matrix so that its first row is +1 and its first column (apart from the first entry) is 1. It is conjectured that skew-hadamard matrices of all orders divisible by 4 exist. The smallest unsolved case is 188. Doubly regular tournaments If we delete the first row and column of a skew-hadamard matrix, and replace the diagonal 1s by 0s, we obtain the adjacency matrix of a doubly regular tournament. This means a tournament on n = 4t + 3 vertices, in which each vertex has inand out-degree 2t + 1, and for any two distinct vertices v and w, there are t vertices z with v z and w z. Conversely, any doubly regular tournament on n vertices gives a skew-hadamard matrix on n + 1 vertices. In a forthcoming paper, Bailey, Cameron, Filipiak, Kunert and Markiewicz use Hamiltonian decompositions of doubly regular tournaments to construct universally optimal circular repeated-measurements designs. Problem Does every doubly regular tournament have a Hamiltonian decomposition? Indeed, Kelly conjectured in the 1960s that every regular tournament has a Hamiltonian decomposition. An example This is related to the Fano plane: Paley tournaments The simplest construction of doubly regular tournaments starts with a finite field of order q 3 (mod 4). The vertices are the elements of the field, and there is an arc x y if and only if y x is a square. (This is a tournament because 1 is a non-square, and therefore y x is a square if and only if x y is not.) If q is prime, then there is an obvious Hamiltonian decomposition: for each non-zero square s, take the Hamiltonian cycle (0, s, 2s, 3s,..., s). However, if q is not a prime, it is not so obvious how to proceed. Conference matrices A conference matrix of order n is an n n matrix C with diagonal entries 0 and off-diagonal entries ±1 which satisfies CC = (n 1)I. We have: The defining equation shows that any two rows of C are orthogonal. The contributions to the inner product of the ith and jth rows coming from the ith and jth positions are zero; each further position contributes +1 or 1; there must be equally many (namely (n 2)/2) contributions of each sign. So n is even. The defining equation gives C 1 = (1/(n 1))C, whence C C = (n 1)I. So the columns are also pairwise orthogonal. The property of being a conference matrix is unchanged under changing the sign of any row or column, or simultaneously applying the same permutation to rows and columns. Symmetric and skew-symmetric Using row and column sign changes, we can assume that all entries in the first row and column (apart from their intersection) are +1; then any row other than the first has n/2 entries +1 (including the first entry) and (n 2)/2 entries 1. Let C be such a matrix, and let S be the matrix obtained from C by deleting the first row and column. If n 2 (mod 4) then S is symmetric; if n 0 (mod 4) then S is skew-symmetric.

4 Proof of the theorem Suppose first that S is not symmetric. Without loss of generality, we can assume that S 12 = +1 while S 21 = 1. Each row of S has m entries +1 and m entries 1, where n = 2m + 2; and the inner product of two rows is 1. Suppose that the first two rows look as follows: a b c d Now row 1 gives a + b = m 1, c + d = m; row 2 gives a + c = m, b + d = m 1; and the inner product gives a + d = m 1, b + c = m. From these we obtain By slight abuse of language, we call a normalised conference matrix C symmetric or skew according as S is symmetric or skew (that is, according to the congruence on n (mod 4)). A symmetric conference matrix really is symmetric, while a skew conference matrix becomes skew if we change the sign of the first column. a = 1 2 ((a + b) + (a + c) (b + c)) = (m 1)/2, so m is odd, and n 0 (mod 4). The other case is similar. Symmetric conference matrices Let C be a symmetric conference matrix. Let A be obtained from S by replacing +1 by 0 and 1 by 1.Then A is the incidence matrix of a strongly regular graph of Paley type: that is, a graph with n 1 vertices in which every vertex has degree (n 2)/2, two adjacent vertices have (n 6)/4 common neighbours, and two non-adjacent vertices have (n 2)/4 common neighbours. The matrix S is called the Seidel adjacency matrix of the graph. The complementary graph has the same properties. Symmetric conference matrices are associated with other combinatorial objects, among them regular two-graphs, sets of equiangular lines in Euclidean space, switching classes of graphs. A conference matrix can produce many different strongly regular graphs by choosing different rows and columns for the normalisation. Again the Paley construction works, on a field of order q +1 (mod 4); join x to y if y x is a square. (This time, 1 is a square, so y x is a square if and only if x y is.) An example The Paley graph on 5 vertices is the 5-cycle. We obtain a symmetric conference matrix by bordering the Seidel adjacency matrix as shown Another example A new first row and column, with 0 in the (1, 1) position and other entries +, gives a symmetric conference matrix of order 10. The MSG logo is the Paley graph on GF(9). (Exercise: Prove this!) A theorem of van Lint and Seidel asserts that, if a symmetric conference matrix of order n exists, then n 1 is the sum of two squares. Thus there is no such matrix of order 22 or 34. They exist for all other orders up to 42 which are congruent to 2 (mod 4), and a complete classification of these is known up to order 30. The simplest construction is that by Paley, in the case where n 1 is a prime power: the matrix S has rows and columns indexed by the finite field of order n 1, and the (i, j) entry is +1 if j i is a non-zero square in the field, 1 if it is a non-square, and 0 if i = j. Symmetric conference matrices first arose in the field of conference telephony.

5 Skew conference matrices Let C be a skew conference matrix. By changing the sign of the first column, we can ensure that C really is skew: that is, C = C. Now (C + I)(C + I) = ni, so H = C + I is a Hadamard matrix. It is a skew-hadamard matrix, as defined earlier; apart from the diagonal, it is skew. Conversely, if H is a skew-hadamard matrix, then H I is a skew conference matrix. If C is a skew conference matrix, then S is the adjacency matrix of a doubly regular tournament, as we saw earlier. (Recall that this is a directed graph on n 1 vertices in which every vertex has in-degree and out-degree (n 2)/2 and every pair of vertices have (n 4)/4 common in-neighbours (and the same number of out-neighbours). Again this is equivalent to the existence of a skew conference matrix. Dennis Lin s problem Dennis Lin is interested in skew-symmetric matrices C with diagonal entries 0 (as they must be) and off-diagonal entries ±1, and also in matrices of the form H = C + I with C as described. He is interested in the largest possible determinant of such matrices of given size. Of course, it is natural to use the letters C and H for such matrices, but they are not necessarily conference or Hadamard matrices. So I will call them cold matrices and hot matrices respectively. Of course, if n is a multiple of 4, the maximum determinant for C is realised by a skew conference matrix (if one exists, as is conjectured to be always the case), and the maximum determinant for H is realised by a skew-hadamard matrix. In other words, the maximum-determinant cold and hot matrices C and H are related by H = C + I. In view of the skew-hadamard conjecture, I will not consider multiples of 4 for which a skew conference matrix fails to exist. A skew-symmetric matrix of odd order has determinant zero; so there is nothing interesting to say in this case. So the remaining case is that in which n is congruent to 2 (mod 4). Lin made the first half of the following conjecture, and the second half seems as well supported: Conjecture For orders congruent to 2 (mod 4), if C is a cold matrix with maximum determinant, then C + I is a hot matrix with maximum determinant; and, if H is a hot matrix with maximum determinant, then H I is a cold matrix with maximum determinant. Of course, he is also interested in the related questions: What is the maximum determinant? How do you construct matrices achieving this maximum (or at least coming close)? Hot matrices Meeting the Ehlich Wojtas bound Ehlich and Wojtas (independently) considered the question of the largest possible determinant of a matrix with entries ±1 when the order is not a multiple of 4. They showed: For n 2 (mod 4), the determinant of an n n matrix with entries ±1 is at most 2(n 1)(n 2) (n 2)/2. Of course this is also an upper bound for the determinant of a hot matrix. We believe there should be a similar bound for the determinant of a cold matrix. Will Orrick (personal communication) showed: A hot matrix of order n can achieve the Ehlich Wojtas bound if and only if 2n 3 is a perfect square. This allows n = 6, 14, 26 and 42, but forbids, for example, n = 10, 18 and 22.

6 Computational results These are due to me, Will Orrick, and Gordon Royle. Lin s conjecture is confirmed for n = 6 and n = 10. The maximum determinants of hot and cold matrices are (160, 81) for n = 6 (the former meeting the EW bound) and (64000, 33489) for n = 10 (the EW bound is 73728). In each case there is a unique maximising matrix up to equivalence. Random search by Gordon Royle gives strong evidence for the truth of Lin s conjecture for n = 14, 18, 22 and 26, and indeed finds only a few equivalence classes of maximising matrices in these cases. Will Orrick searched larger matrices, assuming a special bi-circulant form for the matrices. He was less convinced of the truth of Lin s conjecture; he conjectures that the maximum determinant of a hot matrix is at least cn n/2 for some positive constant c, and found pairs of hot matrices with determinants around 0.45n n/2 where the determinants of the corresponding cold matrices are ordered the other way.

SHARP 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.,

Recall that two vectors in are perpendicular or orthogonal provided that their dot

Orthogonal Complements and Projections Recall that two vectors in are perpendicular or orthogonal provided that their dot product vanishes That is, if and only if Example 1 The vectors in are orthogonal

The 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 degree-diameter problem for circulant graphs of degree 8 and 9 Journal Article How to cite:

α = u v. In other words, Orthogonal Projection

Orthogonal Projection Given any nonzero vector v, it is possible to decompose an arbitrary vector u into a component that points in the direction of v and one that points in a direction orthogonal to v

= 2 + 1 2 2 = 3 4, Now assume that P (k) is true for some fixed k 2. This means that

Instructions. Answer each of the questions on your own paper, and be sure to show your work so that partial credit can be adequately assessed. Credit will not be given for answers (even correct ones) without

Classification 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

Similar matrices and Jordan form

Similar matrices and Jordan form We ve nearly covered the entire heart of linear algebra once we ve finished singular value decompositions we ll have seen all the most central topics. A T A is positive

Notes on Factoring. MA 206 Kurt Bryan

The General Approach Notes on Factoring MA 26 Kurt Bryan Suppose I hand you n, a 2 digit integer and tell you that n is composite, with smallest prime factor around 5 digits. Finding a nontrivial factor

Labeling outerplanar graphs with maximum degree three

Labeling outerplanar graphs with maximum degree three Xiangwen Li 1 and Sanming Zhou 2 1 Department of Mathematics Huazhong Normal University, Wuhan 430079, China 2 Department of Mathematics and Statistics

3. INNER PRODUCT SPACES

. INNER PRODUCT SPACES.. Definition So far we have studied abstract vector spaces. These are a generalisation of the geometric spaces R and R. But these have more structure than just that of a vector space.

(67902) Topics in Theory and Complexity Nov 2, 2006. Lecture 7

(67902) Topics in Theory and Complexity Nov 2, 2006 Lecturer: Irit Dinur Lecture 7 Scribe: Rani Lekach 1 Lecture overview This Lecture consists of two parts In the first part we will refresh the definition

Chapter 7. Permutation Groups

Chapter 7 Permutation Groups () We started the study of groups by considering planar isometries In the previous chapter, we learnt that finite groups of planar isometries can only be cyclic or dihedral

Cameron Liebler line classes

Cameron Liebler line classes Alexander Gavrilyuk Krasovsky Institute of Mathematics and Mechanics, (Yekaterinburg, Russia) based on joint work with Ivan Mogilnykh, Institute of Mathematics (Novosibirsk,

Inner Product Spaces and Orthogonality

Inner Product Spaces and Orthogonality week 3-4 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,

Mathematics Course 111: Algebra I Part IV: Vector Spaces

Mathematics Course 111: Algebra I Part IV: Vector Spaces D. R. Wilkins Academic Year 1996-7 9 Vector Spaces A vector space over some field K is an algebraic structure consisting of a set V on which are

Two classes of ternary codes and their weight distributions

Two classes of ternary codes and their weight distributions Cunsheng Ding, Torleiv Kløve, and Francesco Sica Abstract In this paper we describe two classes of ternary codes, determine their minimum weight

Eigenvalues, Eigenvectors, Matrix Factoring, and Principal Components

Eigenvalues, Eigenvectors, Matrix Factoring, and Principal Components The eigenvalues and eigenvectors of a square matrix play a key role in some important operations in statistics. In particular, they

Numerical Analysis Lecture Notes

Numerical Analysis Lecture Notes Peter J. Olver 5. Inner Products and Norms The norm of a vector is a measure of its size. Besides the familiar Euclidean norm based on the dot product, there are a number

Linear Codes. Chapter 3. 3.1 Basics

Chapter 3 Linear Codes In order to define codes that we can encode and decode efficiently, we add more structure to the codespace. We shall be mainly interested in linear codes. A linear code of length

Orthogonal Projections

Orthogonal Projections and Reflections (with exercises) by D. Klain Version.. Corrections and comments are welcome! Orthogonal Projections Let X,..., X k be a family of linearly independent (column) vectors

26. Determinants I. 1. Prehistory

26. Determinants I 26.1 Prehistory 26.2 Definitions 26.3 Uniqueness and other properties 26.4 Existence Both as a careful review of a more pedestrian viewpoint, and as a transition to a coordinate-independent

3. Linear Programming and Polyhedral Combinatorics

Massachusetts Institute of Technology Handout 6 18.433: Combinatorial Optimization February 20th, 2009 Michel X. Goemans 3. Linear Programming and Polyhedral Combinatorics Summary of what was seen in the

Manifold Learning Examples PCA, LLE and ISOMAP

Manifold Learning Examples PCA, LLE and ISOMAP Dan Ventura October 14, 28 Abstract We try to give a helpful concrete example that demonstrates how to use PCA, LLE and Isomap, attempts to provide some intuition

x1 x 2 x 3 y 1 y 2 y 3 x 1 y 2 x 2 y 1 0.

Cross product 1 Chapter 7 Cross product We are getting ready to study integration in several variables. Until now we have been doing only differential calculus. One outcome of this study will be our ability

Math 215 HW #6 Solutions

Math 5 HW #6 Solutions Problem 34 Show that x y is orthogonal to x + y if and only if x = y Proof First, suppose x y is orthogonal to x + y Then since x, y = y, x In other words, = x y, x + y = (x y) T

ON INDUCED SUBGRAPHS WITH ALL DEGREES ODD. 1. Introduction

ON INDUCED SUBGRAPHS WITH ALL DEGREES ODD A.D. SCOTT Abstract. Gallai proved that the vertex set of any graph can be partitioned into two sets, each inducing a subgraph with all degrees even. We prove

All trees contain a large induced subgraph having all degrees 1 (mod k)

All trees contain a large induced subgraph having all degrees 1 (mod k) David M. Berman, A.J. Radcliffe, A.D. Scott, Hong Wang, and Larry Wargo *Department of Mathematics University of New Orleans New

1 VECTOR SPACES AND SUBSPACES

1 VECTOR SPACES AND SUBSPACES What is a vector? Many are familiar with the concept of a vector as: Something which has magnitude and direction. an ordered pair or triple. a description for quantities such

Revised Version of Chapter 23. We learned long ago how to solve linear congruences. ax c (mod m)

Chapter 23 Squares Modulo p Revised Version of Chapter 23 We learned long ago how to solve linear congruences ax c (mod m) (see Chapter 8). It s now time to take the plunge and move on to quadratic equations.

Math 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)(

Lecture 15 An Arithmetic Circuit Lowerbound and Flows in Graphs

CSE599s: Extremal Combinatorics November 21, 2011 Lecture 15 An Arithmetic Circuit Lowerbound and Flows in Graphs Lecturer: Anup Rao 1 An Arithmetic Circuit Lower Bound An arithmetic circuit is just like

Graphs without proper subgraphs of minimum degree 3 and short cycles

Graphs without proper subgraphs of minimum degree 3 and short cycles Lothar Narins, Alexey Pokrovskiy, Tibor Szabó Department of Mathematics, Freie Universität, Berlin, Germany. August 22, 2014 Abstract

Section 1.1. Introduction to R n

The Calculus of Functions of Several Variables Section. Introduction to R n Calculus is the study of functional relationships and how related quantities change with each other. In your first exposure to

On Integer Additive Set-Indexers of Graphs

On Integer Additive Set-Indexers of Graphs arxiv:1312.7672v4 [math.co] 2 Mar 2014 N K Sudev and K A Germina Abstract A set-indexer of a graph G is an injective set-valued function f : V (G) 2 X such that

(Refer Slide Time: 01.26)

Discrete Mathematical Structures Dr. Kamala Krithivasan Department of Computer Science and Engineering Indian Institute of Technology, Madras Lecture # 27 Pigeonhole Principle In the next few lectures

Math 312 Homework 1 Solutions

Math 31 Homework 1 Solutions Last modified: July 15, 01 This homework is due on Thursday, July 1th, 01 at 1:10pm Please turn it in during class, or in my mailbox in the main math office (next to 4W1) Please

U.C. Berkeley CS276: Cryptography Handout 0.1 Luca Trevisan January, 2009. Notes on Algebra

U.C. Berkeley CS276: Cryptography Handout 0.1 Luca Trevisan January, 2009 Notes on Algebra These notes contain as little theory as possible, and most results are stated without proof. Any introductory

ISOMETRIES OF R n KEITH CONRAD

ISOMETRIES OF R n KEITH CONRAD 1. Introduction An isometry of R n is a function h: R n R n that preserves the distance between vectors: h(v) h(w) = v w for all v and w in R n, where (x 1,..., x n ) = x

The Singular Value Decomposition in Symmetric (Löwdin) Orthogonalization and Data Compression

The Singular Value Decomposition in Symmetric (Löwdin) Orthogonalization and Data Compression The SVD is the most generally applicable of the orthogonal-diagonal-orthogonal type matrix decompositions Every

Lecture 14: Section 3.3

Lecture 14: Section 3.3 Shuanglin Shao October 23, 2013 Definition. Two nonzero vectors u and v in R n are said to be orthogonal (or perpendicular) if u v = 0. We will also agree that the zero vector in

ERDOS PROBLEMS ON IRREGULARITIES OF LINE SIZES AND POINT DEGREES A. GYARFAS*

BOLYAI SOCIETY MATHEMATICAL STUDIES, 11 Paul Erdos and his Mathematics. II, Budapest, 2002, pp. 367-373. ERDOS PROBLEMS ON IRREGULARITIES OF LINE SIZES AND POINT DEGREES A. GYARFAS* Problems and results

Elements of Abstract Group Theory

Chapter 2 Elements of Abstract Group Theory Mathematics is a game played according to certain simple rules with meaningless marks on paper. David Hilbert The importance of symmetry in physics, and for

Problem Set 5 Due: In class Thursday, Oct. 18 Late papers will be accepted until 1:00 PM Friday.

Math 312, Fall 2012 Jerry L. Kazdan Problem Set 5 Due: In class Thursday, Oct. 18 Late papers will be accepted until 1:00 PM Friday. In addition to the problems below, you should also know how to solve

Recall the basic property of the transpose (for any A): v A t Aw = v w, v, w R n.

ORTHOGONAL MATRICES Informally, an orthogonal n n matrix is the n-dimensional analogue of the rotation matrices R θ in R 2. When does a linear transformation of R 3 (or R n ) deserve to be called a rotation?

The Clar Structure of Fullerenes

The Clar Structure of Fullerenes Liz Hartung Massachusetts College of Liberal Arts June 12, 2013 Liz Hartung (Massachusetts College of Liberal Arts) The Clar Structure of Fullerenes June 12, 2013 1 / 25

THE NUMBER OF GRAPHS AND A RANDOM GRAPH WITH A GIVEN DEGREE SEQUENCE. Alexander Barvinok

THE NUMBER OF GRAPHS AND A RANDOM GRAPH WITH A GIVEN DEGREE SEQUENCE Alexer Barvinok Papers are available at http://www.math.lsa.umich.edu/ barvinok/papers.html This is a joint work with J.A. Hartigan

LINEAR ALGEBRA. September 23, 2010

LINEAR ALGEBRA September 3, 00 Contents 0. LU-decomposition.................................... 0. Inverses and Transposes................................. 0.3 Column Spaces and NullSpaces.............................

A Second Course in Mathematics Concepts for Elementary Teachers: Theory, Problems, and Solutions

A Second Course in Mathematics Concepts for Elementary Teachers: Theory, Problems, and Solutions Marcel B. Finan Arkansas Tech University c All Rights Reserved First Draft February 8, 2006 1 Contents 25

Cryptography and Network Security. Prof. D. Mukhopadhyay. Department of Computer Science and Engineering. Indian Institute of Technology, Kharagpur

Cryptography and Network Security Prof. D. Mukhopadhyay Department of Computer Science and Engineering Indian Institute of Technology, Kharagpur Module No. # 01 Lecture No. # 12 Block Cipher Standards

December 4, 2013 MATH 171 BASIC LINEAR ALGEBRA B. KITCHENS

December 4, 2013 MATH 171 BASIC LINEAR ALGEBRA B KITCHENS The equation 1 Lines in two-dimensional space (1) 2x y = 3 describes a line in two-dimensional space The coefficients of x and y in the equation

State of Stress at Point

State of Stress at Point Einstein Notation The basic idea of Einstein notation is that a covector and a vector can form a scalar: This is typically written as an explicit sum: According to this convention,

3. Let A and B be two n n orthogonal matrices. Then prove that AB and BA are both orthogonal matrices. Prove a similar result for unitary matrices.

Exercise 1 1. Let A be an n n orthogonal matrix. Then prove that (a) the rows of A form an orthonormal basis of R n. (b) the columns of A form an orthonormal basis of R n. (c) for any two vectors x,y R

AN ALGORITHM FOR DETERMINING WHETHER A GIVEN BINARY MATROID IS GRAPHIC

AN ALGORITHM FOR DETERMINING WHETHER A GIVEN BINARY MATROID IS GRAPHIC W. T. TUTTE. Introduction. In a recent series of papers [l-4] on graphs and matroids I used definitions equivalent to the following.

1 Sets and Set Notation.

LINEAR ALGEBRA MATH 27.6 SPRING 23 (COHEN) LECTURE NOTES Sets and Set Notation. Definition (Naive Definition of a Set). A set is any collection of objects, called the elements of that set. We will most

The degree, size and chromatic index of a uniform hypergraph

The degree, size and chromatic index of a uniform hypergraph Noga Alon Jeong Han Kim Abstract Let H be a k-uniform hypergraph in which no two edges share more than t common vertices, and let D denote the

Section 5.3. Section 5.3. u m ] l jj. = l jj u j + + l mj u m. v j = [ u 1 u j. l mj

Section 5. l j v j = [ u u j u m ] l jj = l jj u j + + l mj u m. l mj Section 5. 5.. Not orthogonal, the column vectors fail to be perpendicular to each other. 5..2 his matrix is orthogonal. Check that

Split 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

Group Theory. Contents

Group Theory Contents Chapter 1: Review... 2 Chapter 2: Permutation Groups and Group Actions... 3 Orbits and Transitivity... 6 Specific Actions The Right regular and coset actions... 8 The Conjugation

A permutation can also be represented by describing its cycles. What do you suppose is meant by this?

Shuffling, Cycles, and Matrices Warm up problem. Eight people stand in a line. From left to right their positions are numbered,,,... 8. The eight people then change places according to THE RULE which directs

CS 5614: (Big) Data Management Systems. B. Aditya Prakash Lecture #18: Dimensionality Reduc7on

CS 5614: (Big) Data Management Systems B. Aditya Prakash Lecture #18: Dimensionality Reduc7on Dimensionality Reduc=on Assump=on: Data lies on or near a low d- dimensional subspace Axes of this subspace

COUNTING INDEPENDENT SETS IN SOME CLASSES OF (ALMOST) REGULAR GRAPHS

COUNTING INDEPENDENT SETS IN SOME CLASSES OF (ALMOST) REGULAR GRAPHS Alexander Burstein Department of Mathematics Howard University Washington, DC 259, USA aburstein@howard.edu Sergey Kitaev Mathematics

A Linear Online 4-DSS Algorithm

A Linear Online 4-DSS Algorithm V. Kovalevsky 1990: algorithm K1990 We consider the recognition of 4-DSSs on the frontier of a region in the 2D cellular grid. Algorithm K1990 is one of the simplest (implementation)

Row Ideals and Fibers of Morphisms

Michigan Math. J. 57 (2008) Row Ideals and Fibers of Morphisms David Eisenbud & Bernd Ulrich Affectionately dedicated to Mel Hochster, who has been an inspiration to us for many years, on the occasion

LINEAR ALGEBRA W W L CHEN

LINEAR ALGEBRA W W L CHEN c W W L Chen, 1997, 2008 This chapter is available free to all individuals, on understanding that it is not to be used for financial gain, and may be downloaded and/or photocopied,

ABSTRACT. For example, circle orders are the containment orders of circles (actually disks) in the plane (see [8,9]).

Degrees of Freedom Versus Dimension for Containment Orders Noga Alon 1 Department of Mathematics Tel Aviv University Ramat Aviv 69978, Israel Edward R. Scheinerman 2 Department of Mathematical Sciences

1 Chapter 13. VECTORS IN THREE DIMENSIONAL SPACE Let s begin with some names and notation for things: R is the set (collection) of real numbers. We write x R to mean that x is a real number. A real number

USE 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

11 Ideals. 11.1 Revisiting Z

11 Ideals The presentation here is somewhat different than the text. In particular, the sections do not match up. We have seen issues with the failure of unique factorization already, e.g., Z[ 5] = O Q(

A Turán Type Problem Concerning the Powers of the Degrees of a Graph

A Turán Type Problem Concerning the Powers of the Degrees of a Graph Yair Caro and Raphael Yuster Department of Mathematics University of Haifa-ORANIM, Tivon 36006, Israel. AMS Subject Classification:

Direct Methods for Solving Linear Systems. Matrix Factorization

Direct Methods for Solving Linear Systems Matrix Factorization Numerical Analysis (9th Edition) R L Burden & J D Faires Beamer Presentation Slides prepared by John Carroll Dublin City University c 2011

Design of LDPC codes

Design of LDPC codes Codes from finite geometries Random codes: Determine the connections of the bipartite Tanner graph by using a (pseudo)random algorithm observing the degree distribution of the code

Factoring. Factoring 1

Factoring Factoring 1 Factoring Security of RSA algorithm depends on (presumed) difficulty of factoring o Given N = pq, find p or q and RSA is broken o Rabin cipher also based on factoring Factoring like

Factoring Algorithms

Factoring Algorithms The p 1 Method and Quadratic Sieve November 17, 2008 () Factoring Algorithms November 17, 2008 1 / 12 Fermat s factoring method Fermat made the observation that if n has two factors

Large induced subgraphs with all degrees odd

Large induced subgraphs with all degrees odd A.D. Scott Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, England Abstract: We prove that every connected graph of order

A network flow algorithm for reconstructing. binary images from discrete X-rays

A network flow algorithm for reconstructing binary images from discrete X-rays Kees Joost Batenburg Leiden University and CWI, The Netherlands kbatenbu@math.leidenuniv.nl Abstract We present a new algorithm

Approximation Algorithms

Approximation Algorithms or: How I Learned to Stop Worrying and Deal with NP-Completeness Ong Jit Sheng, Jonathan (A0073924B) March, 2012 Overview Key Results (I) General techniques: Greedy algorithms

12.5 Equations of Lines and Planes

Instructor: Longfei Li Math 43 Lecture Notes.5 Equations of Lines and Planes What do we need to determine a line? D: a point on the line: P 0 (x 0, y 0 ) direction (slope): k 3D: a point on the line: P

Integer Factorization using the Quadratic Sieve

Integer Factorization using the Quadratic Sieve Chad Seibert* Division of Science and Mathematics University of Minnesota, Morris Morris, MN 56567 seib0060@morris.umn.edu March 16, 2011 Abstract We give

ON THE DEGREES OF FREEDOM OF SIGNALS ON GRAPHS. Mikhail Tsitsvero and Sergio Barbarossa

ON THE DEGREES OF FREEDOM OF SIGNALS ON GRAPHS Mikhail Tsitsvero and Sergio Barbarossa Sapienza Univ. of Rome, DIET Dept., Via Eudossiana 18, 00184 Rome, Italy E-mail: tsitsvero@gmail.com, sergio.barbarossa@uniroma1.it

Linear algebra and the geometry of quadratic equations. Similarity transformations and orthogonal matrices

MATH 30 Differential Equations Spring 006 Linear algebra and the geometry of quadratic equations Similarity transformations and orthogonal matrices First, some things to recall from linear algebra Two

1.3. DOT PRODUCT 19. 6. If θ is the angle (between 0 and π) between two non-zero vectors u and v,

1.3. DOT PRODUCT 19 1.3 Dot Product 1.3.1 Definitions and Properties The dot product is the first way to multiply two vectors. The definition we will give below may appear arbitrary. But it is not. It

6. Cholesky factorization

6. Cholesky factorization EE103 (Fall 2011-12) triangular matrices forward and backward substitution the Cholesky factorization solving Ax = b with A positive definite inverse of a positive definite matrix

MATH1231 Algebra, 2015 Chapter 7: Linear maps

MATH1231 Algebra, 2015 Chapter 7: Linear maps A/Prof. Daniel Chan School of Mathematics and Statistics University of New South Wales danielc@unsw.edu.au Daniel Chan (UNSW) MATH1231 Algebra 1 / 43 Chapter

Dimensionality Reduction: Principal Components Analysis

Dimensionality Reduction: Principal Components Analysis In data mining one often encounters situations where there are a large number of variables in the database. In such situations it is very likely

CSC2420 Fall 2012: Algorithm Design, Analysis and Theory

CSC2420 Fall 2012: Algorithm Design, Analysis and Theory Allan Borodin November 15, 2012; Lecture 10 1 / 27 Randomized online bipartite matching and the adwords problem. We briefly return to online algorithms

Degree Hypergroupoids Associated with Hypergraphs

Filomat 8:1 (014), 119 19 DOI 10.98/FIL1401119F Published by Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: http://www.pmf.ni.ac.rs/filomat Degree Hypergroupoids Associated

Identifying second degree equations

Chapter 7 Identifing second degree equations 7.1 The eigenvalue method In this section we appl eigenvalue methods to determine the geometrical nature of the second degree equation a 2 + 2h + b 2 + 2g +

Ranking on Data Manifolds

Ranking on Data Manifolds Dengyong Zhou, Jason Weston, Arthur Gretton, Olivier Bousquet, and Bernhard Schölkopf Max Planck Institute for Biological Cybernetics, 72076 Tuebingen, Germany {firstname.secondname

HOLES 5.1. INTRODUCTION

HOLES 5.1. INTRODUCTION One of the major open problems in the field of art gallery theorems is to establish a theorem for polygons with holes. A polygon with holes is a polygon P enclosing several other

Component Ordering in Independent Component Analysis Based on Data Power

Component Ordering in Independent Component Analysis Based on Data Power Anne Hendrikse Raymond Veldhuis University of Twente University of Twente Fac. EEMCS, Signals and Systems Group Fac. EEMCS, Signals

Sum of squares of degrees in a graph

Sum of squares of degrees in a graph Bernardo M. Ábrego 1 Silvia Fernández-Merchant 1 Michael G. Neubauer William Watkins Department of Mathematics California State University, Northridge 18111 Nordhoff

CONTROLLABILITY. Chapter 2. 2.1 Reachable Set and Controllability. Suppose we have a linear system described by the state equation

Chapter 2 CONTROLLABILITY 2 Reachable Set and Controllability Suppose we have a linear system described by the state equation ẋ Ax + Bu (2) x() x Consider the following problem For a given vector x in

WHICH SCORING RULE MAXIMIZES CONDORCET EFFICIENCY? 1. Introduction

WHICH SCORING RULE MAXIMIZES CONDORCET EFFICIENCY? DAVIDE P. CERVONE, WILLIAM V. GEHRLEIN, AND WILLIAM S. ZWICKER Abstract. Consider an election in which each of the n voters casts a vote consisting of

NCSS Statistical Software Principal Components Regression. In ordinary least squares, the regression coefficients are estimated using the formula ( )

Chapter 340 Principal Components Regression Introduction is a technique for analyzing multiple regression data that suffer from multicollinearity. When multicollinearity occurs, least squares estimates

Math 231b Lecture 35. G. Quick

Math 231b Lecture 35 G. Quick 35. Lecture 35: Sphere bundles and the Adams conjecture 35.1. Sphere bundles. Let X be a connected finite cell complex. We saw that the J-homomorphism could be defined by

Solutions to old Exam 1 problems

Solutions to old Exam 1 problems Hi students! I am putting this old version of my review for the first midterm review, place and time to be announced. Check for updates on the web site as to which sections

SALEM COMMUNITY COLLEGE Carneys Point, New Jersey 08069 COURSE SYLLABUS COVER SHEET. Action Taken (Please Check One) New Course Initiated

SALEM COMMUNITY COLLEGE Carneys Point, New Jersey 08069 COURSE SYLLABUS COVER SHEET Course Title Course Number Department Linear Algebra Mathematics MAT-240 Action Taken (Please Check One) New Course Initiated

MATHEMATICAL ENGINEERING TECHNICAL REPORTS. An Improved Approximation Algorithm for the Traveling Tournament Problem

MATHEMATICAL ENGINEERING TECHNICAL REPORTS An Improved Approximation Algorithm for the Traveling Tournament Problem Daisuke YAMAGUCHI, Shinji IMAHORI, Ryuhei MIYASHIRO, Tomomi MATSUI METR 2009 42 September