Linear and Geometric Algebra October 2020 Printing
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
Geometric Algebra for Vector Fields Analysis and Visualization: Mathematical Settings, Overview and Applications Chantal Oberson Ausoni, Pascal Frey
Geometric algebra for vector fields analysis and visualization: mathematical settings, overview and applications Chantal Oberson Ausoni, Pascal Frey To cite this version: Chantal Oberson Ausoni, Pascal Frey. Geometric algebra for vector fields analysis and visualization: mathematical settings, overview and applications. 2014. hal-00920544v2 HAL Id: hal-00920544 https://hal.sorbonne-universite.fr/hal-00920544v2 Preprint submitted on 18 Sep 2014 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Geometric algebra for vector field analysis and visualization: mathematical settings, overview and applications Chantal Oberson Ausoni and Pascal Frey Abstract The formal language of Clifford’s algebras is attracting an increasingly large community of mathematicians, physicists and software developers seduced by the conciseness and the efficiency of this compelling system of mathematics. This contribution will suggest how these concepts can be used to serve the purpose of scientific visualization and more specifically to reveal the general structure of complex vector fields. We will emphasize the elegance and the ubiquitous nature of the geometric algebra approach, as well as point out the computational issues at stake. 1 Introduction Nowadays, complex numerical simulations (e.g. in climate modelling, weather fore- cast, aeronautics, genomics, etc.) produce very large data sets, often several ter- abytes, that become almost impossible to process in a reasonable amount of time. -
Exploring Physics with Geometric Algebra, Book II., , C December 2016 COPYRIGHT
peeter joot [email protected] EXPLORINGPHYSICSWITHGEOMETRICALGEBRA,BOOKII. EXPLORINGPHYSICSWITHGEOMETRICALGEBRA,BOOKII. peeter joot [email protected] December 2016 – version v.1.3 Peeter Joot [email protected]: Exploring physics with Geometric Algebra, Book II., , c December 2016 COPYRIGHT Copyright c 2016 Peeter Joot All Rights Reserved This book may be reproduced and distributed in whole or in part, without fee, subject to the following conditions: • The copyright notice above and this permission notice must be preserved complete on all complete or partial copies. • Any translation or derived work must be approved by the author in writing before distri- bution. • If you distribute this work in part, instructions for obtaining the complete version of this document must be included, and a means for obtaining a complete version provided. • Small portions may be reproduced as illustrations for reviews or quotes in other works without this permission notice if proper citation is given. Exceptions to these rules may be granted for academic purposes: Write to the author and ask. Disclaimer: I confess to violating somebody’s copyright when I copied this copyright state- ment. v DOCUMENTVERSION Version 0.6465 Sources for this notes compilation can be found in the github repository https://github.com/peeterjoot/physicsplay The last commit (Dec/5/2016), associated with this pdf was 595cc0ba1748328b765c9dea0767b85311a26b3d vii Dedicated to: Aurora and Lance, my awesome kids, and Sofia, who not only tolerates and encourages my studies, but is also awesome enough to think that math is sexy. PREFACE This is an exploratory collection of notes containing worked examples of more advanced appli- cations of Geometric Algebra (GA), also known as Clifford Algebra. -
Arxiv:2003.06292V1 [Math.GR] 12 Mar 2020 Eggnrtr N Ignlmti.Tedaoa Arxi Matrix Diagonal the Matrix
ALGORITHMS IN LINEAR ALGEBRAIC GROUPS SUSHIL BHUNIA, AYAN MAHALANOBIS, PRALHAD SHINDE, AND ANUPAM SINGH ABSTRACT. This paper presents some algorithms in linear algebraic groups. These algorithms solve the word problem and compute the spinor norm for orthogonal groups. This gives us an algorithmic definition of the spinor norm. We compute the double coset decompositionwith respect to a Siegel maximal parabolic subgroup, which is important in computing infinite-dimensional representations for some algebraic groups. 1. INTRODUCTION Spinor norm was first defined by Dieudonné and Kneser using Clifford algebras. Wall [21] defined the spinor norm using bilinear forms. These days, to compute the spinor norm, one uses the definition of Wall. In this paper, we develop a new definition of the spinor norm for split and twisted orthogonal groups. Our definition of the spinornorm is rich in the sense, that itis algorithmic in nature. Now one can compute spinor norm using a Gaussian elimination algorithm that we develop in this paper. This paper can be seen as an extension of our earlier work in the book chapter [3], where we described Gaussian elimination algorithms for orthogonal and symplectic groups in the context of public key cryptography. In computational group theory, one always looks for algorithms to solve the word problem. For a group G defined by a set of generators hXi = G, the problem is to write g ∈ G as a word in X: we say that this is the word problem for G (for details, see [18, Section 1.4]). Brooksbank [4] and Costi [10] developed algorithms similar to ours for classical groups over finite fields. -
Orthogonal Reduction 1 the Row Echelon Form -.: Mathematical
MATH 5330: Computational Methods of Linear Algebra Lecture 9: Orthogonal Reduction Xianyi Zeng Department of Mathematical Sciences, UTEP 1 The Row Echelon Form Our target is to solve the normal equation: AtAx = Atb ; (1.1) m×n where A 2 R is arbitrary; we have shown previously that this is equivalent to the least squares problem: min jjAx−bjj : (1.2) x2Rn t n×n As A A2R is symmetric positive semi-definite, we can try to compute the Cholesky decom- t t n×n position such that A A = L L for some lower-triangular matrix L 2 R . One problem with this approach is that we're not fully exploring our information, particularly in Cholesky decomposition we treat AtA as a single entity in ignorance of the information about A itself. t m×m In particular, the structure A A motivates us to study a factorization A=QE, where Q2R m×n is orthogonal and E 2 R is to be determined. Then we may transform the normal equation to: EtEx = EtQtb ; (1.3) t m×m where the identity Q Q = Im (the identity matrix in R ) is used. This normal equation is equivalent to the least squares problem with E: t min Ex−Q b : (1.4) x2Rn Because orthogonal transformation preserves the L2-norm, (1.2) and (1.4) are equivalent to each n other. Indeed, for any x 2 R : jjAx−bjj2 = (b−Ax)t(b−Ax) = (b−QEx)t(b−QEx) = [Q(Qtb−Ex)]t[Q(Qtb−Ex)] t t t t t t t t 2 = (Q b−Ex) Q Q(Q b−Ex) = (Q b−Ex) (Q b−Ex) = Ex−Q b : Hence the target is to find an E such that (1.3) is easier to solve. -
Multivector Differentiation and Linear Algebra 0.5Cm 17Th Santaló
Multivector differentiation and Linear Algebra 17th Santalo´ Summer School 2016, Santander Joan Lasenby Signal Processing Group, Engineering Department, Cambridge, UK and Trinity College Cambridge [email protected], www-sigproc.eng.cam.ac.uk/ s jl 23 August 2016 1 / 78 Examples of differentiation wrt multivectors. Linear Algebra: matrices and tensors as linear functions mapping between elements of the algebra. Functional Differentiation: very briefly... Summary Overview The Multivector Derivative. 2 / 78 Linear Algebra: matrices and tensors as linear functions mapping between elements of the algebra. Functional Differentiation: very briefly... Summary Overview The Multivector Derivative. Examples of differentiation wrt multivectors. 3 / 78 Functional Differentiation: very briefly... Summary Overview The Multivector Derivative. Examples of differentiation wrt multivectors. Linear Algebra: matrices and tensors as linear functions mapping between elements of the algebra. 4 / 78 Summary Overview The Multivector Derivative. Examples of differentiation wrt multivectors. Linear Algebra: matrices and tensors as linear functions mapping between elements of the algebra. Functional Differentiation: very briefly... 5 / 78 Overview The Multivector Derivative. Examples of differentiation wrt multivectors. Linear Algebra: matrices and tensors as linear functions mapping between elements of the algebra. Functional Differentiation: very briefly... Summary 6 / 78 We now want to generalise this idea to enable us to find the derivative of F(X), in the A ‘direction’ – where X is a general mixed grade multivector (so F(X) is a general multivector valued function of X). Let us use ∗ to denote taking the scalar part, ie P ∗ Q ≡ hPQi. Then, provided A has same grades as X, it makes sense to define: F(X + tA) − F(X) A ∗ ¶XF(X) = lim t!0 t The Multivector Derivative Recall our definition of the directional derivative in the a direction F(x + ea) − F(x) a·r F(x) = lim e!0 e 7 / 78 Let us use ∗ to denote taking the scalar part, ie P ∗ Q ≡ hPQi. -
Geometric-Algebra Adaptive Filters Wilder B
1 Geometric-Algebra Adaptive Filters Wilder B. Lopes∗, Member, IEEE, Cassio G. Lopesy, Senior Member, IEEE Abstract—This paper presents a new class of adaptive filters, namely Geometric-Algebra Adaptive Filters (GAAFs). They are Faces generated by formulating the underlying minimization problem (a deterministic cost function) from the perspective of Geometric Algebra (GA), a comprehensive mathematical language well- Edges suited for the description of geometric transformations. Also, (directed lines) differently from standard adaptive-filtering theory, Geometric Calculus (the extension of GA to differential calculus) allows Fig. 1. A polyhedron (3-dimensional polytope) can be completely described for applying the same derivation techniques regardless of the by the geometric multiplication of its edges (oriented lines, vectors), which type (subalgebra) of the data, i.e., real, complex numbers, generate the faces and hypersurfaces (in the case of a general n-dimensional quaternions, etc. Relying on those characteristics (among others), polytope). a deterministic quadratic cost function is posed, from which the GAAFs are devised, providing a generalization of regular adaptive filters to subalgebras of GA. From the obtained update rule, it is shown how to recover the following least-mean squares perform calculus with hypercomplex quantities, i.e., elements (LMS) adaptive filter variants: real-entries LMS, complex LMS, that generalize complex numbers for higher dimensions [2]– and quaternions LMS. Mean-square analysis and simulations in [10]. a system identification scenario are provided, showing very good agreement for different levels of measurement noise. GA-based AFs were first introduced in [11], [12], where they were successfully employed to estimate the geometric Index Terms—Adaptive filtering, geometric algebra, quater- transformation (rotation and translation) that aligns a pair of nions. -
Determinants in Geometric Algebra
Determinants in Geometric Algebra Eckhard Hitzer 16 June 2003, recovered+expanded May 2020 1 Definition Let f be a linear map1, of a real linear vector space Rn into itself, an endomor- phism n 0 n f : a 2 R ! a 2 R : (1) This map is extended by outermorphism (symbol f) to act linearly on multi- vectors f(a1 ^ a2 ::: ^ ak) = f(a1) ^ f(a2) ::: ^ f(ak); k ≤ n: (2) By definition f is grade-preserving and linear, mapping multivectors to mul- tivectors. Examples are the reflections, rotations and translations described earlier. The outermorphism of a product of two linear maps fg is the product of the outermorphisms f g f[g(a1)] ^ f[g(a2)] ::: ^ f[g(ak)] = f[g(a1) ^ g(a2) ::: ^ g(ak)] = f[g(a1 ^ a2 ::: ^ ak)]; (3) with k ≤ n. The square brackets can safely be omitted. The n{grade pseudoscalars of a geometric algebra are unique up to a scalar factor. This can be used to define the determinant2 of a linear map as det(f) = f(I)I−1 = f(I) ∗ I−1; and therefore f(I) = det(f)I: (4) For an orthonormal basis fe1; e2;:::; eng the unit pseudoscalar is I = e1e2 ::: en −1 q q n(n−1)=2 with inverse I = (−1) enen−1 ::: e1 = (−1) (−1) I, where q gives the number of basis vectors, that square to −1 (the linear space is then Rp;q). According to Grassmann n-grade vectors represent oriented volume elements of dimension n. The determinant therefore shows how these volumes change under linear maps. -
Implementation of Gaussian- Elimination
International Journal of Innovative Technology and Exploring Engineering (IJITEE) ISSN: 2278-3075, Volume-5 Issue-11, April 2016 Implementation of Gaussian- Elimination Awatif M.A. Elsiddieg Abstract: Gaussian elimination is an algorithm for solving systems of linear equations, can also use to find the rank of any II. PIVOTING matrix ,we use Gaussian Jordan elimination to find the inverse of a non singular square matrix. This work gives basic concepts The objective of pivoting is to make an element above or in section (1) , show what is pivoting , and implementation of below a leading one into a zero. Gaussian elimination to solve a system of linear equations. The "pivot" or "pivot element" is an element on the left Section (2) we find the rank of any matrix. Section (3) we use hand side of a matrix that you want the elements above and Gaussian elimination to find the inverse of a non singular square matrix. We compare the method by Gauss Jordan method. In below to be zero. Normally, this element is a one. If you can section (4) practical implementation of the method we inherit the find a book that mentions pivoting, they will usually tell you computation features of Gaussian elimination we use programs that you must pivot on a one. If you restrict yourself to the in Matlab software. three elementary row operations, then this is a true Keywords: Gaussian elimination, algorithm Gauss, statement. However, if you are willing to combine the Jordan, method, computation, features, programs in Matlab, second and third elementary row operations, you come up software. -
Naïve Gaussian Elimination Jamie Trahan, Autar Kaw, Kevin Martin University of South Florida United States of America [email protected]
nbm_sle_sim_naivegauss.nb 1 Naïve Gaussian Elimination Jamie Trahan, Autar Kaw, Kevin Martin University of South Florida United States of America [email protected] Introduction One of the most popular numerical techniques for solving simultaneous linear equations is Naïve Gaussian Elimination method. The approach is designed to solve a set of n equations with n unknowns, [A][X]=[C], where A nxn is a square coefficient matrix, X nx1 is the solution vector, and C nx1 is the right hand side array. Naïve Gauss consists of two steps: 1) Forward Elimination: In this step, the unknown is eliminated in each equation starting with the first@ D equation. This way, the equations @areD "reduced" to one equation and@ oneD unknown in each equation. 2) Back Substitution: In this step, starting from the last equation, each of the unknowns is found. To learn more about Naïve Gauss Elimination as well as the pitfall's of the method, click here. A simulation of Naive Gauss Method follows. Section 1: Input Data Below are the input parameters to begin the simulation. This is the only section that requires user input. Once the values are entered, Mathematica will calculate the solution vector [X]. èNumber of equations, n: n = 4 4 è nxn coefficient matrix, [A]: nbm_sle_sim_naivegauss.nb 2 A = Table 1, 10, 100, 1000 , 1, 15, 225, 3375 , 1, 20, 400, 8000 , 1, 22.5, 506.25, 11391 ;A MatrixForm 1 10@88 100 1000 < 8 < 18 15 225 3375< 8 <<D êê 1 20 400 8000 ji 1 22.5 506.25 11391 zy j z j z j z j z è nx1 rightj hand side array, [RHS]: z k { RHS = Table 227.04, 362.78, 517.35, 602.97 ; RHS MatrixForm 227.04 362.78 @8 <D êê 517.35 ji 602.97 zy j z j z j z j z j z k { Section 2: Naïve Gaussian Elimination Method The following sections divide Naïve Gauss elimination into two steps: 1) Forward Elimination 2) Back Substitution To conduct Naïve Gauss Elimination, Mathematica will join the [A] and [RHS] matrices into one augmented matrix, [C], that will facilitate the process of forward elimination. -
Finite Projective Geometries 243
FINITE PROJECTÎVEGEOMETRIES* BY OSWALD VEBLEN and W. H. BUSSEY By means of such a generalized conception of geometry as is inevitably suggested by the recent and wide-spread researches in the foundations of that science, there is given in § 1 a definition of a class of tactical configurations which includes many well known configurations as well as many new ones. In § 2 there is developed a method for the construction of these configurations which is proved to furnish all configurations that satisfy the definition. In §§ 4-8 the configurations are shown to have a geometrical theory identical in most of its general theorems with ordinary projective geometry and thus to afford a treatment of finite linear group theory analogous to the ordinary theory of collineations. In § 9 reference is made to other definitions of some of the configurations included in the class defined in § 1. § 1. Synthetic definition. By a finite projective geometry is meant a set of elements which, for sugges- tiveness, are called points, subject to the following five conditions : I. The set contains a finite number ( > 2 ) of points. It contains subsets called lines, each of which contains at least three points. II. If A and B are distinct points, there is one and only one line that contains A and B. HI. If A, B, C are non-collinear points and if a line I contains a point D of the line AB and a point E of the line BC, but does not contain A, B, or C, then the line I contains a point F of the line CA (Fig. -
New Foundations for Geometric Algebra1
Text published in the electronic journal Clifford Analysis, Clifford Algebras and their Applications vol. 2, No. 3 (2013) pp. 193-211 New foundations for geometric algebra1 Ramon González Calvet Institut Pere Calders, Campus Universitat Autònoma de Barcelona, 08193 Cerdanyola del Vallès, Spain E-mail : [email protected] Abstract. New foundations for geometric algebra are proposed based upon the existing isomorphisms between geometric and matrix algebras. Each geometric algebra always has a faithful real matrix representation with a periodicity of 8. On the other hand, each matrix algebra is always embedded in a geometric algebra of a convenient dimension. The geometric product is also isomorphic to the matrix product, and many vector transformations such as rotations, axial symmetries and Lorentz transformations can be written in a form isomorphic to a similarity transformation of matrices. We collect the idea Dirac applied to develop the relativistic electron equation when he took a basis of matrices for the geometric algebra instead of a basis of geometric vectors. Of course, this way of understanding the geometric algebra requires new definitions: the geometric vector space is defined as the algebraic subspace that generates the rest of the matrix algebra by addition and multiplication; isometries are simply defined as the similarity transformations of matrices as shown above, and finally the norm of any element of the geometric algebra is defined as the nth root of the determinant of its representative matrix of order n. The main idea of this proposal is an arithmetic point of view consisting of reversing the roles of matrix and geometric algebras in the sense that geometric algebra is a way of accessing, working and understanding the most fundamental conception of matrix algebra as the algebra of transformations of multiple quantities. -
Notices of the American Mathematical Society 35 Monticello Place, Pawtucket, RI 02861 USA American Mathematical Society Distribution Center
ISSN 0002-9920 Notices of the American Mathematical Society 35 Monticello Place, Pawtucket, RI 02861 USA Society Distribution Center American Mathematical of the American Mathematical Society February 2012 Volume 59, Number 2 Conformal Mappings in Geometric Algebra Page 264 Infl uential Mathematicians: Birth, Education, and Affi liation Page 274 Supporting the Next Generation of “Stewards” in Mathematics Education Page 288 Lawrence Meeting Page 352 Volume 59, Number 2, Pages 257–360, February 2012 About the Cover: MathJax (see page 346) Trim: 8.25" x 10.75" 104 pages on 40 lb Velocity • Spine: 1/8" • Print Cover on 9pt Carolina AMS-Simons Travel Grants TS AN GR EL The AMS is accepting applications for the second RAV T year of the AMS-Simons Travel Grants program. Each grant provides an early career mathematician with $2,000 per year for two years to reimburse travel expenses related to research. Sixty new awards will be made in 2012. Individuals who are not more than four years past the comple- tion of the PhD are eligible. The department of the awardee will also receive a small amount of funding to help enhance its research atmosphere. The deadline for 2012 applications is March 30, 2012. Applicants must be located in the United States or be U.S. citizens. For complete details of eligibility and application instructions, visit: www.ams.org/programs/travel-grants/AMS-SimonsTG Solve the differential equation. t ln t dr + r = 7tet dt 7et + C r = ln t WHO HAS THE #1 HOMEWORK SYSTEM FOR CALCULUS? THE ANSWER IS IN THE QUESTIONS.