The Square-Root Isometry of Coupled Quadratic Spaces
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Quaternions and Cli Ord Geometric Algebras
Quaternions and Cliord Geometric Algebras Robert Benjamin Easter First Draft Edition (v1) (c) copyright 2015, Robert Benjamin Easter, all rights reserved. Preface As a rst rough draft that has been put together very quickly, this book is likely to contain errata and disorganization. The references list and inline citations are very incompete, so the reader should search around for more references. I do not claim to be the inventor of any of the mathematics found here. However, some parts of this book may be considered new in some sense and were in small parts my own original research. Much of the contents was originally written by me as contributions to a web encyclopedia project just for fun, but for various reasons was inappropriate in an encyclopedic volume. I did not originally intend to write this book. This is not a dissertation, nor did its development receive any funding or proper peer review. I oer this free book to the public, such as it is, in the hope it could be helpful to an interested reader. June 19, 2015 - Robert B. Easter. (v1) [email protected] 3 Table of contents Preface . 3 List of gures . 9 1 Quaternion Algebra . 11 1.1 The Quaternion Formula . 11 1.2 The Scalar and Vector Parts . 15 1.3 The Quaternion Product . 16 1.4 The Dot Product . 16 1.5 The Cross Product . 17 1.6 Conjugates . 18 1.7 Tensor or Magnitude . 20 1.8 Versors . 20 1.9 Biradials . 22 1.10 Quaternion Identities . 23 1.11 The Biradial b/a . -
Codimension One Isometric Immersions Betweenlorentz Spaces by L
TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 252, August 1979 CODIMENSION ONE ISOMETRIC IMMERSIONS BETWEENLORENTZ SPACES BY L. K. GRAVES In memory of my father, Lucius Kingman Graves Abstract. The theorem of Hartman and Nirenberg classifies codimension one isometric immersions between Euclidean spaces as cylinders over plane curves. Corresponding results are given here for Lorentz spaces, which are Euclidean spaces with one negative-definite direction (also known as Minkowski spaces). The pivotal result involves the completeness of the relative nullity foliation of such an immersion. When this foliation carries a nondegenerate metric, results analogous to the Hartman-Nirenberg theorem obtain. Otherwise, a new description, based on particular surfaces in the three-dimensional Lorentz space, is required. The theorem of Hartman and Nirenberg [HN] says that, up to a proper motion of E"+', all isometric immersions of E" into E"+ ' have the form id X c: E"-1 xEUr'x E2 (0.1) where c: E1 ->E2 is a unit speed plane curve and the factors in the product are orthogonal. A proof of the Hartman-Nirenberg result also appears in [N,]. The major step in proving the theorem is showing that the relative nullity foliation, which is spanned by those tangent directions in which a local unit normal field is parallel, has complete leaves. These leaves yield the E"_1 factors in (0.1). The one-dimensional complement of the relative nullity foliation gives rise to the curve c. This paper studies isometric immersions of L" into Ln+1, where L" denotes the «-dimensional Lorentz (or Minkowski) space. Its chief goal is the classifi- cation, up to a proper motion of L"+1, of all such immersions. -
Embeddings of Integral Quadratic Forms Rick Miranda Colorado State
Embeddings of Integral Quadratic Forms Rick Miranda Colorado State University David R. Morrison University of California, Santa Barbara Copyright c 2009, Rick Miranda and David R. Morrison Preface The authors ran a seminar on Integral Quadratic Forms at the Institute for Advanced Study in the Spring of 1982, and worked on a book-length manuscript reporting on the topic throughout the 1980’s and early 1990’s. Some new results which are proved in the manuscript were announced in two brief papers in the Proceedings of the Japan Academy of Sciences in 1985 and 1986. We are making this preliminary version of the manuscript available at this time in the hope that it will be useful. Still to do before the manuscript is in final form: final editing of some portions, completion of the bibliography, and the addition of a chapter on the application to K3 surfaces. Rick Miranda David R. Morrison Fort Collins and Santa Barbara November, 2009 iii Contents Preface iii Chapter I. Quadratic Forms and Orthogonal Groups 1 1. Symmetric Bilinear Forms 1 2. Quadratic Forms 2 3. Quadratic Modules 4 4. Torsion Forms over Integral Domains 7 5. Orthogonality and Splitting 9 6. Homomorphisms 11 7. Examples 13 8. Change of Rings 22 9. Isometries 25 10. The Spinor Norm 29 11. Sign Structures and Orientations 31 Chapter II. Quadratic Forms over Integral Domains 35 1. Torsion Modules over a Principal Ideal Domain 35 2. The Functors ρk 37 3. The Discriminant of a Torsion Bilinear Form 40 4. The Discriminant of a Torsion Quadratic Form 45 5. -
Newton's Method for the Matrix Square Root*
MATHEMATICS OF COMPUTATION VOLUME 46, NUMBER 174 APRIL 1986, PAGES 537-549 Newton's Method for the Matrix Square Root* By Nicholas J. Higham Abstract. One approach to computing a square root of a matrix A is to apply Newton's method to the quadratic matrix equation F( X) = X2 - A =0. Two widely-quoted matrix square root iterations obtained by rewriting this Newton iteration are shown to have excellent mathematical convergence properties. However, by means of a perturbation analysis and supportive numerical examples, it is shown that these simplified iterations are numerically unstable. A further variant of Newton's method for the matrix square root, recently proposed in the literature, is shown to be, for practical purposes, numerically stable. 1. Introduction. A square root of an n X n matrix A with complex elements, A e C"x", is a solution X e C"*" of the quadratic matrix equation (1.1) F(X) = X2-A=0. A natural approach to computing a square root of A is to apply Newton's method to (1.1). For a general function G: CXn -* Cx", Newton's method for the solution of G(X) = 0 is specified by an initial approximation X0 and the recurrence (see [14, p. 140], for example) (1.2) Xk+l = Xk-G'{XkylG{Xk), fc = 0,1,2,..., where G' denotes the Fréchet derivative of G. Identifying F(X+ H) = X2 - A +(XH + HX) + H2 with the Taylor series for F we see that F'(X) is a linear operator, F'(X): Cx" ^ C"x", defined by F'(X)H= XH+ HX. -
Sensitivity and Stability Analysis of Nonlinear Kalman Filters with Application to Aircraft Attitude Estimation
Graduate Theses, Dissertations, and Problem Reports 2013 Sensitivity and stability analysis of nonlinear Kalman filters with application to aircraft attitude estimation Matthew Brandon Rhudy West Virginia University Follow this and additional works at: https://researchrepository.wvu.edu/etd Recommended Citation Rhudy, Matthew Brandon, "Sensitivity and stability analysis of nonlinear Kalman filters with application ot aircraft attitude estimation" (2013). Graduate Theses, Dissertations, and Problem Reports. 3659. https://researchrepository.wvu.edu/etd/3659 This Dissertation is protected by copyright and/or related rights. It has been brought to you by the The Research Repository @ WVU with permission from the rights-holder(s). You are free to use this Dissertation in any way that is permitted by the copyright and related rights legislation that applies to your use. For other uses you must obtain permission from the rights-holder(s) directly, unless additional rights are indicated by a Creative Commons license in the record and/ or on the work itself. This Dissertation has been accepted for inclusion in WVU Graduate Theses, Dissertations, and Problem Reports collection by an authorized administrator of The Research Repository @ WVU. For more information, please contact [email protected]. SENSITIVITY AND STABILITY ANALYSIS OF NONLINEAR KALMAN FILTERS WITH APPLICATION TO AIRCRAFT ATTITUDE ESTIMATION by Matthew Brandon Rhudy Dissertation submitted to the Benjamin M. Statler College of Engineering and Mineral Resources at West Virginia University in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Aerospace Engineering Approved by Dr. Yu Gu, Committee Chairperson Dr. John Christian Dr. Gary Morris Dr. Marcello Napolitano Dr. Powsiri Klinkhachorn Department of Mechanical and Aerospace Engineering Morgantown, West Virginia 2013 Keywords: Attitude Estimation, Extended Kalman Filter, GPS/INS Sensor Fusion, Stochastic Stability Copyright 2013, Matthew B. -
Matlib: Matrix Functions for Teaching and Learning Linear Algebra and Multivariate Statistics
Package ‘matlib’ August 21, 2021 Type Package Title Matrix Functions for Teaching and Learning Linear Algebra and Multivariate Statistics Version 0.9.5 Date 2021-08-10 Maintainer Michael Friendly <[email protected]> Description A collection of matrix functions for teaching and learning matrix linear algebra as used in multivariate statistical methods. These functions are mainly for tutorial purposes in learning matrix algebra ideas using R. In some cases, functions are provided for concepts available elsewhere in R, but where the function call or name is not obvious. In other cases, functions are provided to show or demonstrate an algorithm. In addition, a collection of functions are provided for drawing vector diagrams in 2D and 3D. License GPL (>= 2) Language en-US URL https://github.com/friendly/matlib BugReports https://github.com/friendly/matlib/issues LazyData TRUE Suggests knitr, rglwidget, rmarkdown, carData, webshot2, markdown Additional_repositories https://dmurdoch.github.io/drat Imports xtable, MASS, rgl, car, methods VignetteBuilder knitr RoxygenNote 7.1.1 Encoding UTF-8 NeedsCompilation no Author Michael Friendly [aut, cre] (<https://orcid.org/0000-0002-3237-0941>), John Fox [aut], Phil Chalmers [aut], Georges Monette [ctb], Gaston Sanchez [ctb] 1 2 R topics documented: Repository CRAN Date/Publication 2021-08-21 15:40:02 UTC R topics documented: adjoint . .3 angle . .4 arc..............................................5 arrows3d . .6 buildTmat . .8 cholesky . .9 circle3d . 10 class . 11 cofactor . 11 cone3d . 12 corner . 13 Det.............................................. 14 echelon . 15 Eigen . 16 gaussianElimination . 17 Ginv............................................. 18 GramSchmidt . 20 gsorth . 21 Inverse............................................ 22 J............................................... 23 len.............................................. 23 LU.............................................. 24 matlib . 25 matrix2latex . 27 minor . 27 MoorePenrose . -
Introduction to Clifford's Geometric Algebra
SICE Journal of Control, Measurement, and System Integration, Vol. 4, No. 1, pp. 001–011, January 2011 Introduction to Clifford’s Geometric Algebra ∗ Eckhard HITZER Abstract : Geometric algebra was initiated by W.K. Clifford over 130 years ago. It unifies all branchesof physics, and has found rich applications in robotics, signal processing, ray tracing, virtual reality, computer vision, vector field processing, tracking, geographic information systems and neural computing. This tutorial explains the basics of geometric algebra, with concrete examples of the plane, of 3D space, of spacetime, and the popular conformal model. Geometric algebras are ideal to represent geometric transformations in the general framework of Clifford groups (also called versor or Lipschitz groups). Geometric (algebra based) calculus allows, e.g., to optimize learning algorithms of Clifford neurons, etc. Key Words: Hypercomplex algebra, hypercomplex analysis, geometry, science, engineering. 1. Introduction unique associative and multilinear algebra with this property. W.K. Clifford (1845-1879), a young English Goldsmid pro- Thus if we wish to generalize methods from algebra, analysis, ff fessor of applied mathematics at the University College of Lon- calculus, di erential geometry (etc.) of real numbers, complex don, published in 1878 in the American Journal of Mathemat- numbers, and quaternion algebra to vector spaces and multivec- ics Pure and Applied a nine page long paper on Applications of tor spaces (which include additional elements representing 2D Grassmann’s Extensive Algebra. In this paper, the young ge- up to nD subspaces, i.e. plane elements up to hypervolume el- ff nius Clifford, standing on the shoulders of two giants of alge- ements), the study of Cli ord algebras becomes unavoidable. -
Notes on Linear Algebra and Matrix Analysis
Notes on Linear Algebra and Matrix Analysis Maxim Neumann May 2006, Version 0.1.1 1 Matrix Basics Literature to this topic: [1–4]. x†y ⇐⇒ < y,x >: standard inner product. x†x = 1 : x is normalized x†y = 0 : x,y are orthogonal x†y = 0, x†x = 1, y†y = 1 : x,y are orthonormal Ax = y is uniquely solvable if A is linear independent (nonsingular). Majorization: Arrange b and a in increasing order (bm,am) then: k k b majorizes a ⇐⇒ ∑ bmi ≥ ∑ ami ∀ k ∈ [1,...,n] (1) i=1 i=1 n n The collection of all vectors b ∈ R that majorize a given vector a ∈ R may be obtained by forming the convex hull of n! vectors, which are computed by permuting the n components of a. Direct sum of matrices A ∈ Mn1,B ∈ Mn2: A 0 A ⊕ B = ∈ M (2) 0 B n1+n2 [A,B] = traceAB†: matrix inner product. 1.1 Trace n traceA = ∑λi (3) i trace(A + B) = traceA + traceB (4) traceAB = traceBA (5) 1.2 Determinants The determinant det(A) expresses the volume of a matrix A. A is singular. Linear equation is not solvable. det(A) = 0 ⇐⇒ −1 (6) A does not exists vectors in A are linear dependent det(A) 6= 0 ⇐⇒ A is regular/nonsingular. Ai j ∈ R → det(A) ∈ R Ai j ∈ C → det(A) ∈ C If A is a square matrix(An×n) and has the eigenvalues λi, then det(A) = ∏λi detAT = detA (7) detA† = detA (8) detAB = detA detB (9) Elementary operations on matrix and determinant: Interchange of two rows : detA ∗ = −1 Multiplication of a row by a nonzero scalar c : detA ∗ = c Addition of a scalar multiple of one row to another row : detA = detA 1 2 EIGENVALUES, EIGENVECTORS, AND SIMILARITY 2 a b = ad − bc (10) c d 2 Eigenvalues, Eigenvectors, and Similarity σ(An×n) = {λ1,...,λn} is the set of eigenvalues of A, also called the spectrum of A. -
Zeros of Conformal Fields in Any Metric Signature
Zeros of conformal fields in any metric signature Andrzej Derdzinski Department of Mathematics, The Ohio State University, Columbus, OH 43210, USA E-mail: [email protected] Abstract. The connected components of the zero set of any conformal vector field, in a pseudo-Riemannian manifold of arbitrary signature, are shown to be totally umbilical conifold varieties, that is, smooth submanifolds except possibly for some quadric singularities. The singularities occur only when the metric is indefinite, including the Lorentzian case. This generalizes an analogous result for Riemannian manifolds, due to Belgun, Moroianu and Ornea (2010). AMS classification scheme numbers: 53B30 1. Introduction A vector field v on a pseudo-Riemannian manifold (M; g) of dimension n ≥ 2 is called conformal if, for some function φ : M ! IR, $vg = φg; that is; in coordinates; vj;k + vk;j = φgjk : (1) One then obviously has div v = nφ/2. The class of conformal vector fields on (M; g) includes Killing fields v, characterized by (1) with φ = 0. Kobayashi [15] showed that, for any Killing vector field v on a Riemannian manifold (M; g), the connected components of the zero set of v are mutually isolated totally geodesic submanifolds of even codimensions. Assuming compactness of M, Blair [5] established an analogue of Kobayashi's theorem for conformal vector fields, in which the word `geodesic' is replaced by `umbilical' and the codimension clause is relaxed for one-point connected components. Very recently, Belgun, Moroianu and Ornea [4] proved that Blair's conclusion is valid in the noncompact case as well. The last result is also a direct consequence of a theorem of Frances [12] stating that a conformal field on a Riemannian manifold is linearizable at any zero z unless some neighborhood of z is conformally flat. -
Functions Preserving Matrix Groups and Iterations for the Matrix Square Root∗
FUNCTIONS PRESERVING MATRIX GROUPS AND ITERATIONS FOR THE MATRIX SQUARE ROOT∗ NICHOLAS J. HIGHAM† , D. STEVEN MACKEY‡ , NILOUFER MACKEY§ , AND FRANC¸OISE TISSEUR¶ Abstract. For which functions f does A ∈ G ⇒ f(A) ∈ G when G is the matrix automorphism group associated with a bilinear or sesquilinear form? For example, if A is symplectic when is f(A) symplectic? We show that group structure is preserved precisely when f(A−1)= f(A)−1 for bilinear forms and when f(A−∗) = f(A)−∗ for sesquilinear forms. Meromorphic functions that satisfy each of these conditions are characterized. Related to structure preservation is the condition f(A)= f(A), and analytic functions and rational functions satisfying this condition are also characterized. These results enable us to characterize all meromorphic functions that map every G into itself as the ratio of a polynomial and its “reversal”, up to a monomial factor and conjugation. The principal square root is an important example of a function that preserves every automor- phism group G. By exploiting the matrix sign function, a new family of coupled iterations for the matrix square root is derived. Some of these iterations preserve every G; all of them are shown, via a novel Fr´echet derivative-based analysis, to be numerically stable. A rewritten form of Newton’s method for the square root of A ∈ G is also derived. Unlike the original method, this new form has good numerical stability properties, and we argue that it is the iterative method of choice for computing A1/2 when A ∈ G. Our tools include a formula for the sign of a certain block 2 × 2 matrix, the generalized polar decomposition along with a wide class of iterations for computing it, and a connection between the generalized polar decomposition of I + A and the square root of A ∈ G. -
Computing Real Square Roots of a Real Matrix* LINEAR ALGEBRA
Computing Real Square Roots of a Real Matrix* Nicholas J. Higham Department of Mathematics University of Manchester Manchester Ml3 9PL, England In memory of James H. WiIkinson Submitted by Hans Schneider ABSTRACT Bjiirck and Hammarling [l] describe a fast, stable Schur method for computing a square root X of a matrix A (X2 = A).We present an extension of their method which enables real arithmetic to be used throughout when computing a real square root of a real matrix. For a nonsingular real matrix A conditions are given for the existence of a real square root, and for the existence of a real square root which is a polynomial in A; thenumber of square roots of the latter type is determined. The conditioning of matrix square roots is investigated, and an algorithm is given for the computation of a well-conditioned square root. 1. INTRODUCTION Given a matrix A, a matrix X for which X2 = A is called a square root of A. Several authors have considered the computation of matrix square roots [3, 4, 9, 10, 15, 161. A particularly attractive method which utilizes the Schur decomposition is described by Bjiirck and Hammarling [l]; in general it requires complex arithmetic. Our main purpose is to show how the method can be extended so as to compute a real square root of a real matrix, if one exists, in real arithmetic. The theory behind the existence of matrix square roots is nontrivial, as can be seen by noting that while the n x n identity matrix has infinitely many square roots for n > 2 (any involutary matrix such as a Householder transformation is a square root), a nonsingular Jordan block has precisely two square roots (this is proved in Corollary 1). -
Notes on Geometry and Spacetime Version 2.7, November 2009
Notes on Geometry and Spacetime Version 2.7, November 2009 David B. Malament Department of Logic and Philosophy of Science University of California, Irvine [email protected] Contents 1 Preface 2 2 Preliminary Mathematics 2 2.1 Vector Spaces . 3 2.2 Affine Spaces . 6 2.3 Metric Affine Spaces . 15 2.4 Euclidean Geometry . 21 3 Minkowskian Geometry and Its Physical Significance 28 3.1 Minkowskian Geometry { Formal Development . 28 3.2 Minkowskian Geometry { Physical Interpretation . 39 3.3 Uniqueness Result for Minkowskian Angular Measure . 51 3.4 Uniqueness Results for the Relative Simultaneity Relation . 56 4 From Minkowskian Geometry to Hyperbolic Plane Geometry 64 4.1 Tarski's Axioms for first-order Euclidean and Hyperbolic Plane Geometry . 64 4.2 The Hyperboloid Model of Hyperbolic Plane Geometry . 69 These notes have not yet been fully de-bugged. Please read them with caution. (Corrections will be much appreciated.) 1 1 Preface The notes that follow bring together a somewhat unusual collection of topics. In section 3, I discuss the foundations of \special relativity". I emphasize the invariant, \geometrical approach" to the theory, and spend a fair bit of time on one special topic: the status of the relative simultaneity relation within the theory. At issue is whether the standard relation, the one picked out by Einstein's \definition" of simultaneity, is conventional in character, or is rather in some significant sense forced on us. Section 2 is preparatory. When the time comes, I take \Minkowski space- time" to be a four-dimensional affine space endowed with a Lorentzian inner product.