Exploring Linear Algebra
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Chapter 7 Block Designs
Chapter 7 Block Designs One simile that solitary shines In the dry desert of a thousand lines. Epilogue to the Satires, ALEXANDER POPE The collection of all subsets of cardinality k of a set with v elements (k < v) has the ³ ´ v¡t property that any subset of t elements, with 0 · t · k, is contained in precisely k¡t subsets of size k. The subsets of size k provide therefore a nice covering for the subsets of a lesser cardinality. Observe that the number of subsets of size k that contain a subset of size t depends only on v; k; and t and not on the specific subset of size t in question. This is the essential defining feature of the structures that we wish to study. The example we just described inspires general interest in producing similar coverings ³ ´ v without using all the k subsets of size k but rather as small a number of them as possi- 1 2 CHAPTER 7. BLOCK DESIGNS ble. The coverings that result are often elegant geometrical configurations, of which the projective and affine planes are examples. These latter configurations form nice coverings only for the subsets of cardinality 2, that is, any two elements are in the same number of these special subsets of size k which we call blocks (or, in certain instances, lines). A collection of subsets of cardinality k, called blocks, with the property that every subset of size t (t · k) is contained in the same number (say ¸) of blocks is called a t-design. We supply the reader with constructions for t-designs with t as high as 5. -
Systems Analysis of Stochastic and Population Balance Models for Chemically Reacting Systems
Systems Analysis of Stochastic and Population Balance Models for Chemically Reacting Systems by Eric Lynn Haseltine A dissertation submitted in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY (Chemical Engineering) at the UNIVERSITY OF WISCONSIN–MADISON 2005 c Copyright by Eric Lynn Haseltine 2005 All Rights Reserved i To Lori and Grace, for their love and support ii Systems Analysis of Stochastic and Population Balance Models for Chemically Reacting Systems Eric Lynn Haseltine Under the supervision of Professor James B. Rawlings At the University of Wisconsin–Madison Chemical reaction models present one method of analyzing complex reaction pathways. Most models of chemical reaction networks employ a traditional, deterministic setting. The short- comings of this traditional framework, namely difficulty in accounting for population het- erogeneity and discrete numbers of reactants, motivate the need for more flexible modeling frameworks such as stochastic and cell population balance models. How to efficiently use models to perform systems-level tasks such as parameter estimation and feedback controller design is important in all frameworks. Consequently, this thesis focuses on three main areas: 1. improving the methods used to simulate and perform systems-level tasks using stochas- tic models, 2. formulating and applying cell population balance models to better account for experi- mental data, and 3. applying moving-horizon estimation to improve state estimates for nonlinear reaction systems. For stochastic models, we have derived and implemented techniques that improve simulation efficiency and perform systems-level tasks using these simulations. For discrete stochastic models, these systems-level tasks rely on approximate, biased sensitivities, whereas continuous models (i.e. -
Hadamard and Conference Matrices
Hadamard and conference matrices Peter J. Cameron December 2011 with input from Dennis Lin, Will Orrick and Gordon Royle Now det(H) is equal to the volume of the n-dimensional parallelepiped spanned by the rows of H. By assumption, each row has Euclidean length at most n1/2, so that det(H) ≤ nn/2; equality holds if and only if I every entry of H is ±1; > I the rows of H are orthogonal, that is, HH = nI. A matrix attaining the bound is a Hadamard matrix. Hadamard's theorem Let H be an n × n matrix, all of whose entries are at most 1 in modulus. How large can det(H) be? A matrix attaining the bound is a Hadamard matrix. Hadamard's theorem Let H be an n × n matrix, all of whose entries are at most 1 in modulus. How large can det(H) be? Now det(H) is equal to the volume of the n-dimensional parallelepiped spanned by the rows of H. By assumption, each row has Euclidean length at most n1/2, so that det(H) ≤ nn/2; equality holds if and only if I every entry of H is ±1; > I the rows of H are orthogonal, that is, HH = nI. Hadamard's theorem Let H be an n × n matrix, all of whose entries are at most 1 in modulus. How large can det(H) be? Now det(H) is equal to the volume of the n-dimensional parallelepiped spanned by the rows of H. By assumption, each row has Euclidean length at most n1/2, so that det(H) ≤ nn/2; equality holds if and only if I every entry of H is ±1; > I the rows of H are orthogonal, that is, HH = nI. -
Some Applications of Hadamard Matrices
ORIGINAL ARTICLE Jennifer Seberry · Beata J Wysocki · Tadeusz A Wysocki On some applications of Hadamard matrices Abstract Modern communications systems are heavily reliant on statistical tech- niques to recover information in the presence of noise and interference. One of the mathematical structures used to achieve this goal is Hadamard matrices. They are used in many different ways and some examples are given. This paper concentrates on code division multiple access systems where Hadamard matrices are used for user separation. Two older techniques from design and analysis of experiments which rely on similar processes are also included. We give a short bibliography (from the thousands produced by a google search) of applications of Hadamard matrices appearing since the paper of Hedayat and Wallis in 1978 and some appli- cations in telecommunications. 1 Introduction Hadamard matrices seem such simple matrix structures: they are square, have entries +1or−1 and have orthogonal row vectors and orthogonal column vectors. Yet they have been actively studied for over 138 years and still have more secrets to be discovered. In this paper we concentrate on engineering and statistical appli- cations especially those in communications systems, digital image processing and orthogonal spreading sequences for direct sequences spread spectrum code division multiple access. 2 Basic definitions and properties A square matrix with elements ±1 and size h, whose distinct row vectors are mutu- ally orthogonal, is referred to as an Hadamard matrix of order h. The smallest examples are: 2 Seberry et al. 111 1 11 − 1 − 1 (1) , , 1 − 11−− − 11− where − denotes −1. Such matrices were first studied by Sylvester (1867) who observed that if H is an Hadamard matrix, then HH H −H is also an Hadamard matrix. -
Perron Spectratopes and the Real Nonnegative Inverse Eigenvalue Problem
Perron Spectratopes and the Real Nonnegative Inverse Eigenvalue Problem a b, Charles R. Johnson , Pietro Paparella ∗ aDepartment of Mathematics, College of William & Mary, Williamsburg, VA 23187-8795, USA bDivision of Engineering and Mathematics, University of Washington Bothell, Bothell, WA 98011-8246, USA Abstract Call an n-by-n invertible matrix S a Perron similarity if there is a real non- 1 scalar diagonal matrix D such that SDS− is entrywise nonnegative. We give two characterizations of Perron similarities and study the polyhedra (S) := n 1 C x R : SDxS− 0;Dx := diag (x) and (S) := x (S): x1 = 1 , whichf 2 we call the Perron≥ spectracone andgPerronP spectratopef 2, respectively. C Theg set of all normalized real spectra of diagonalizable nonnegative matrices may be covered by Perron spectratopes, so that enumerating them is of interest. The Perron spectracone and spectratope of Hadamard matrices are of par- ticular interest and tend to have large volume. For the canonical Hadamard matrix (as well as other matrices), the Perron spectratope coincides with the convex hull of its rows. In addition, we provide a constructive version of a result due to Fiedler ([9, Theorem 2.4]) for Hadamard orders, and a constructive version of [2, Theorem 5.1] for Sule˘ımanova spectra. Keywords: Perron spectracone, Perron spectratope, real nonnegative inverse eigenvalue problem, Hadamard matrix, association scheme, relative gain array 2010 MSC: 15A18, 15B48, 15A29, 05B20, 05E30 1. Introduction The real nonnegative inverse eigenvalue problem (RNIEP) is to determine arXiv:1508.07400v2 [math.RA] 19 Nov 2015 which sets of n real numbers occur as the spectrum of an n-by-n nonnegative matrix. -
Hadamard Matrices Include
Hadamard and conference matrices Peter J. Cameron University of St Andrews & Queen Mary University of London Mathematics Study Group with input from Rosemary Bailey, Katarzyna Filipiak, Joachim Kunert, Dennis Lin, Augustyn Markiewicz, Will Orrick, Gordon Royle and many happy returns . Happy Birthday, MSG!! Happy Birthday, MSG!! and many happy returns . Now det(H) is equal to the volume of the n-dimensional parallelepiped spanned by the rows of H. By assumption, each row has Euclidean length at most n1/2, so that det(H) ≤ nn/2; equality holds if and only if I every entry of H is ±1; > I the rows of H are orthogonal, that is, HH = nI. A matrix attaining the bound is a Hadamard matrix. This is a nice example of a continuous problem whose solution brings us into discrete mathematics. Hadamard's theorem Let H be an n × n matrix, all of whose entries are at most 1 in modulus. How large can det(H) be? A matrix attaining the bound is a Hadamard matrix. This is a nice example of a continuous problem whose solution brings us into discrete mathematics. Hadamard's theorem Let H be an n × n matrix, all of whose entries are at most 1 in modulus. How large can det(H) be? Now det(H) is equal to the volume of the n-dimensional parallelepiped spanned by the rows of H. By assumption, each row has Euclidean length at most n1/2, so that det(H) ≤ nn/2; equality holds if and only if I every entry of H is ±1; > I the rows of H are orthogonal, that is, HH = nI. -
Opengl Transformations
Transformations CS 537 Interactive Computer Graphics Prof. David E. Breen Department of Computer Science E. Angel and D. Shreiner: Interactive Computer Graphics 6E © Addison-Wesley 2012 1 Objectives • Introduce standard transformations - Rotation - Translation - Scaling - Shear • Derive homogeneous coordinate transformation matrices • Learn to build arbitrary transformation matrices from simple transformations E. Angel and D. Shreiner: Interactive Computer Graphics 6E © Addison-Wesley 2012 2 General Transformations A transformation maps points to other points and/or vectors to other vectors v=T(u) Q=T(P) E. Angel and D. Shreiner: Interactive Computer Graphics 6E © Addison-Wesley 2012 3 Affine Transformations • Line preserving • Characteristic of many physically important transformations - Rigid body transformations: rotation, translation - Scaling, shear • Importance in graphics is that we need only transform endpoints of line segments and let implementation draw line segment between the transformed endpoints E. Angel and D. Shreiner: Interactive Computer Graphics 6E © Addison-Wesley 2012 4 Pipeline Implementation T (from application program) frame u T(u) buffer transformation rasterizer v T(v) T(v) v T(v) T(u) u T(u) vertices vertices pixels E. Angel and D. Shreiner: Interactive Computer Graphics 6E © Addison-Wesley 2012 5 Notation We will be working with both coordinate-free representations of transformations and representations within a particular frame P,Q, R: points in an affine space u, v, w: vectors in an affine space α, β, γ: scalars p, q, r: representations of points -array of 4 scalars in homogeneous coordinates u, v, w: representations of vectors -array of 4 scalars in homogeneous coordinates E. Angel and D. Shreiner: Interactive Computer Graphics 6E © Addison-Wesley 2012 6 Translation • Move (translate, displace) a point to a new location P’ d P • Displacement determined by a vector d - Three degrees of freedom - P’= P+d E. -
Matrix Theory
Matrix Theory Xingzhi Zhan +VEHYEXI7XYHMIW MR1EXLIQEXMGW :SPYQI %QIVMGER1EXLIQEXMGEP7SGMIX] Matrix Theory https://doi.org/10.1090//gsm/147 Matrix Theory Xingzhi Zhan Graduate Studies in Mathematics Volume 147 American Mathematical Society Providence, Rhode Island EDITORIAL COMMITTEE David Cox (Chair) Daniel S. Freed Rafe Mazzeo Gigliola Staffilani 2010 Mathematics Subject Classification. Primary 15-01, 15A18, 15A21, 15A60, 15A83, 15A99, 15B35, 05B20, 47A63. For additional information and updates on this book, visit www.ams.org/bookpages/gsm-147 Library of Congress Cataloging-in-Publication Data Zhan, Xingzhi, 1965– Matrix theory / Xingzhi Zhan. pages cm — (Graduate studies in mathematics ; volume 147) Includes bibliographical references and index. ISBN 978-0-8218-9491-0 (alk. paper) 1. Matrices. 2. Algebras, Linear. I. Title. QA188.Z43 2013 512.9434—dc23 2013001353 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294 USA. Requests can also be made by e-mail to [email protected]. c 2013 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. -
On the Classification of Hadamard Matrices of Order 32 1 Introduction
On the classi¯cation of Hadamard matrices of order 32 H. Kharaghania;1 B. Tayfeh-Rezaieb aDepartment of Mathematics and Computer Science, University of Lethbridge, Lethbridge, Alberta, T1K3M4, Canada bSchool of Mathematics, Institute for Research in Fundamental Sciences (IPM), P.O. Box 19395-5746, Tehran, Iran December 5, 2009 Abstract All equivalence classes of Hadamard matrices of order at most 28 have been found by 1994. Order 32 is where a combinatorial explosion occurs on the number of Hadamard matrices. We ¯nd all equivalence classes of Hadamard matrices of order 32 which are of certain types. It turns out that there are exactly 13,680,757 Hadamard matrices of one type and 26,369 such matrices of another type. Based on experience with the classi¯cation of Hadamard matrices of smaller order, it is expected that the number of the remaining two types of these matrices, relative to the total number of Hadamard matrices of order 32, to be insigni¯cant. AMS Subject Classi¯cation: 05B20, 05B05, 05B30. Keywords: Hadamard matrices, classi¯cation of combinatorial objects, isomorph-free gen- eration, orderly algorithm. 1 Introduction A Hadamard matrix of order n is a (¡1; 1) square matrix H of order n such that HHt = nI, where Ht is the transpose of H and I is the identity matrix. It is well known that the order of a Hadamard matrix is 1,2 or a multiple of 4. The old Hadamard conjecture states that the converse also holds, i.e. there is a Hadamard matrix for any order which is divisible by 4. -
Affine Transforms and Matrix Algebra
Chapter 5 Matrix Algebra and Affine Transformations 5.1 The need for multiple coordinate frames So far we have been dealing with cameras, lights and geometry in our scene by specifying everything with respect to a common origin and coordinate system. This turns out to be very limiting. For example, for animation we will want to be able to move the camera and models and render a sequence of images as time advances to create an illusion of motion. We may wish to build models separately and then combine them together to make a scene. We may wish to define one object relative to another { for example we may want to place a hand at the end of an arm. In order to provide this flexibility we need a good mechanism to provide for multiple coordinate systems and for easy transformation between them. We will call these coordinate systems, which we use to represent a scene, coordinate frames. Figure 5.1 shows an example of what we mean. A cylinder has been built in its own modeling coordinate frame and then rotated and translated into the entire scene's coordinate frame. The set of operations providing for all transformations such as these, are known as the affine transforms. The affines include translations and all linear transformations, like scale, rotate, and shear. 29 30CHAPTER 5. MATRIX ALGEBRA AND AFFINE TRANSFORMATIONS m o Cylinder in model Cylinder in world frame O. The coordinates M. cylinder has been rotated and translated. Figure 5.1: Object in model and world coordinate frames. 5.2 Affine transformations Let us first examine the affine transforms in 2D space, where it is easy to illustrate them with diagrams, then later we will look at the affines in 3D. -
Part IA — Vectors and Matrices
Part IA | Vectors and Matrices Based on lectures by N. Peake Notes taken by Dexter Chua Michaelmas 2014 These notes are not endorsed by the lecturers, and I have modified them (often significantly) after lectures. They are nowhere near accurate representations of what was actually lectured, and in particular, all errors are almost surely mine. Complex numbers Review of complex numbers, including complex conjugate, inverse, modulus, argument and Argand diagram. Informal treatment of complex logarithm, n-th roots and complex powers. de Moivre's theorem. [2] Vectors 3 Review of elementary algebra of vectors in R , including scalar product. Brief discussion n n of vectors in R and C ; scalar product and the Cauchy-Schwarz inequality. Concepts of linear span, linear independence, subspaces, basis and dimension. Suffix notation: including summation convention, δij and "ijk. Vector product and triple product: definition and geometrical interpretation. Solution of linear vector equations. Applications of vectors to geometry, including equations of lines, planes and spheres. [5] Matrices Elementary algebra of 3 × 3 matrices, including determinants. Extension to n × n complex matrices. Trace, determinant, non-singular matrices and inverses. Matrices as linear transformations; examples of geometrical actions including rotations, reflections, dilations, shears; kernel and image. [4] Simultaneous linear equations: matrix formulation; existence and uniqueness of solu- tions, geometric interpretation; Gaussian elimination. [3] Symmetric, anti-symmetric, orthogonal, hermitian and unitary matrices. Decomposition of a general matrix into isotropic, symmetric trace-free and antisymmetric parts. [1] Eigenvalues and Eigenvectors Eigenvalues and eigenvectors; geometric significance. [2] Proof that eigenvalues of hermitian matrix are real, and that distinct eigenvalues give an orthogonal basis of eigenvectors. -
Sir Thomas Muir, 1844–1934
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Linear Algebra and its Applications 411 (2005) 3–67 www.elsevier.com/locate/laa Sir Thomas Muir, 1844–1934 Pieter Maritz Department of Mathematics, University of Stellenbosch, Private Bag X1, Matieland 7602, South Africa Received 9 December 2003; accepted 27 December 2004 Available online 21 February 2005 Submitted by W. Barrett Abstract Part I of this paper on Sir Thomas Muir, deals with his life in Scotland and at the Cape of Good Hope. In 1892, Thomas Muir, mathematician and educator, became the third Superin- tendent General of Education in the Cape Colony under British rule. He will be remembered as one of the greatest organisers and reformers in the history of Cape education. Muir found relief from his arduous administrative duties by his investigations in the field of mathematics, and, in particular, of algebra. Most of his more than 320 papers were on determinants and allied subjects. His magnum opus was a five-volume work: The Theory of Determinants in the Historical Order of Development (London, 1890–1930). Muir’s publications will be covered in Part II of this paper. However, a treatment of the contents of Muir’s papers and his vast contribution to the theory of determinants, fall beyond the scope of this paper. © 2005 Elsevier Inc. All rights reserved. Keywords:Thomas Muir’s life and death; Determinants; Hadamard’s inequality; Law of extensible minors; Muir’s mathematical writings Part I: His Life 1. Introduction The aim of this portion of Cape history in the Introduction is to sketch the situation that Thomas Muir was confronted with when he landed at Cape Town in 1892.