Linear Algebra - Part II Projection, Eigendecomposition, SVD
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Geometric Polarimetry − Part I: Spinors and Wave States
1 Geometric Polarimetry Part I: Spinors and − Wave States David Bebbington, Laura Carrea, and Ernst Krogager, Member, IEEE Abstract A new approach to polarization algebra is introduced. It exploits the geometric properties of spinors in order to represent wave states consistently in arbitrary directions in three dimensional space. In this first expository paper of an intended series the basic derivation of the spinorial wave state is seen to be geometrically related to the electromagnetic field tensor in the spatio-temporal Fourier domain. Extracting the polarization state from the electromagnetic field requires the introduction of a new element, which enters linearly into the defining relation. We call this element the phase flag and it is this that keeps track of the polarization reference when the coordinate system is changed and provides a phase origin for both wave components. In this way we are able to identify the sphere of three dimensional unit wave vectors with the Poincar´esphere. Index Terms state of polarization, geometry, covariant and contravariant spinors and tensors, bivectors, phase flag, Poincar´esphere. I. INTRODUCTION arXiv:0804.0745v1 [physics.optics] 4 Apr 2008 The development of applications in radar polarimetry has been vigorous over the past decade, and is anticipated to continue with ambitious new spaceborne remote sensing missions (for example TerraSAR-X [1] and TanDEM-X [2]). As technical capabilities increase, and new application areas open up, innovations in data analysis often result. Because polarization data are This work was supported by the Marie Curie Research Training Network AMPER (Contract number HPRN-CT-2002-00205). D. Bebbington and L. -
Learning Geometric Algebra by Modeling Motions of the Earth and Shadows of Gnomons to Predict Solar Azimuths and Altitudes
Learning Geometric Algebra by Modeling Motions of the Earth and Shadows of Gnomons to Predict Solar Azimuths and Altitudes April 24, 2018 James Smith [email protected] https://mx.linkedin.com/in/james-smith-1b195047 \Our first step in developing an expression for the orientation of \our" gnomon: Diagramming its location at the instant of the 2016 December solstice." Abstract Because the shortage of worked-out examples at introductory levels is an obstacle to widespread adoption of Geometric Algebra (GA), we use GA to calculate Solar azimuths and altitudes as a function of time via the heliocentric model. We begin by representing the Earth's motions in GA terms. Our representation incorporates an estimate of the time at which the Earth would have reached perihelion in 2017 if not affected by the Moon's gravity. Using the geometry of the December 2016 solstice as a starting 1 point, we then employ GA's capacities for handling rotations to determine the orientation of a gnomon at any given latitude and longitude during the period between the December solstices of 2016 and 2017. Subsequently, we derive equations for two angles: that between the Sun's rays and the gnomon's shaft, and that between the gnomon's shadow and the direction \north" as traced on the ground at the gnomon's location. To validate our equations, we convert those angles to Solar azimuths and altitudes for comparison with simulations made by the program Stellarium. As further validation, we analyze our equations algebraically to predict (for example) the precise timings and locations of sunrises, sunsets, and Solar zeniths on the solstices and equinoxes. -
Schaum's Outline of Linear Algebra (4Th Edition)
SCHAUM’S SCHAUM’S outlines outlines Linear Algebra Fourth Edition Seymour Lipschutz, Ph.D. Temple University Marc Lars Lipson, Ph.D. University of Virginia Schaum’s Outline Series New York Chicago San Francisco Lisbon London Madrid Mexico City Milan New Delhi San Juan Seoul Singapore Sydney Toronto Copyright © 2009, 2001, 1991, 1968 by The McGraw-Hill Companies, Inc. All rights reserved. Except as permitted under the United States Copyright Act of 1976, no part of this publication may be reproduced or distributed in any form or by any means, or stored in a database or retrieval system, without the prior writ- ten permission of the publisher. ISBN: 978-0-07-154353-8 MHID: 0-07-154353-8 The material in this eBook also appears in the print version of this title: ISBN: 978-0-07-154352-1, MHID: 0-07-154352-X. All trademarks are trademarks of their respective owners. Rather than put a trademark symbol after every occurrence of a trademarked name, we use names in an editorial fashion only, and to the benefit of the trademark owner, with no intention of infringement of the trademark. Where such designations appear in this book, they have been printed with initial caps. McGraw-Hill eBooks are available at special quantity discounts to use as premiums and sales promotions, or for use in corporate training programs. To contact a representative please e-mail us at [email protected]. TERMS OF USE This is a copyrighted work and The McGraw-Hill Companies, Inc. (“McGraw-Hill”) and its licensors reserve all rights in and to the work. -
A Fast Method for Computing the Inverse of Symmetric Block Arrowhead Matrices
Appl. Math. Inf. Sci. 9, No. 2L, 319-324 (2015) 319 Applied Mathematics & Information Sciences An International Journal http://dx.doi.org/10.12785/amis/092L06 A Fast Method for Computing the Inverse of Symmetric Block Arrowhead Matrices Waldemar Hołubowski1, Dariusz Kurzyk1,∗ and Tomasz Trawi´nski2 1 Institute of Mathematics, Silesian University of Technology, Kaszubska 23, Gliwice 44–100, Poland 2 Mechatronics Division, Silesian University of Technology, Akademicka 10a, Gliwice 44–100, Poland Received: 6 Jul. 2014, Revised: 7 Oct. 2014, Accepted: 8 Oct. 2014 Published online: 1 Apr. 2015 Abstract: We propose an effective method to find the inverse of symmetric block arrowhead matrices which often appear in areas of applied science and engineering such as head-positioning systems of hard disk drives or kinematic chains of industrial robots. Block arrowhead matrices can be considered as generalisation of arrowhead matrices occurring in physical problems and engineering. The proposed method is based on LDLT decomposition and we show that the inversion of the large block arrowhead matrices can be more effective when one uses our method. Numerical results are presented in the examples. Keywords: matrix inversion, block arrowhead matrices, LDLT decomposition, mechatronic systems 1 Introduction thermal and many others. Considered subsystems are highly differentiated, hence formulation of uniform and simple mathematical model describing their static and A square matrix which has entries equal zero except for dynamic states becomes problematic. The process of its main diagonal, a one row and a column, is called the preparing a proper mathematical model is often based on arrowhead matrix. Wide area of applications causes that the formulation of the equations associated with this type of matrices is popular subject of research related Lagrangian formalism [9], which is a convenient way to with mathematics, physics or engineering, such as describe the equations of mechanical, electromechanical computing spectral decomposition [1], solving inverse and other components. -
Structured Factorizations in Scalar Product Spaces
STRUCTURED FACTORIZATIONS IN SCALAR PRODUCT SPACES D. STEVEN MACKEY¤, NILOUFER MACKEYy , AND FRANC»OISE TISSEURz Abstract. Let A belong to an automorphism group, Lie algebra or Jordan algebra of a scalar product. When A is factored, to what extent do the factors inherit structure from A? We answer this question for the principal matrix square root, the matrix sign decomposition, and the polar decomposition. For general A, we give a simple derivation and characterization of a particular generalized polar decomposition, and we relate it to other such decompositions in the literature. Finally, we study eigendecompositions and structured singular value decompositions, considering in particular the structure in eigenvalues, eigenvectors and singular values that persists across a wide range of scalar products. A key feature of our analysis is the identi¯cation of two particular classes of scalar products, termed unitary and orthosymmetric, which serve to unify assumptions for the existence of structured factorizations. A variety of di®erent characterizations of these scalar product classes is given. Key words. automorphism group, Lie group, Lie algebra, Jordan algebra, bilinear form, sesquilinear form, scalar product, inde¯nite inner product, orthosymmetric, adjoint, factorization, symplectic, Hamiltonian, pseudo-orthogonal, polar decomposition, matrix sign function, matrix square root, generalized polar decomposition, eigenvalues, eigenvectors, singular values, structure preservation. AMS subject classi¯cations. 15A18, 15A21, 15A23, 15A57, -
Exercise 7.5
IN SUMMARY Key Idea • A projection of one vector onto another can be either a scalar or a vector. The difference is the vector projection has a direction. A A a a u u O O b NB b N B Scalar projection of a on b Vector projection of a on b Need to Know ! ! ! ! a # b ! • The scalar projection of a on b ϭϭ! 0 a 0 cos u, where u is the angle @ b @ ! ! between a and b. ! ! ! ! ! ! a # b ! a # b ! • The vector projection of a on b ϭ ! b ϭ a ! ! b b @ b @ 2 b # b ! • The direction cosines for OP ϭ 1a, b, c2 are a b cos a ϭ , cos b ϭ , 2 2 2 2 2 2 Va ϩ b ϩ c Va ϩ b ϩ c c cos g ϭ , where a, b, and g are the direction angles 2 2 2 Va ϩ b ϩ c ! between the position vector OP and the positive x-axis, y-axis and z-axis, respectively. Exercise 7.5 PART A ! 1. a. The vector a ϭ 12, 32 is projected onto the x-axis. What is the scalar projection? What is the vector projection? ! b. What are the scalar and vector projections when a is projected onto the y-axis? 2. Explain why it is not possible to obtain! either a scalar! projection or a vector projection when a nonzero vector x is projected on 0. 398 7.5 SCALAR AND VECTOR PROJECTIONS NEL ! ! 3. Consider two nonzero vectors,a and b, that are perpendicular! ! to each other.! Explain why the scalar and vector projections of a on b must! be !0 and 0, respectively. -
MATH 304 Linear Algebra Lecture 24: Scalar Product. Vectors: Geometric Approach
MATH 304 Linear Algebra Lecture 24: Scalar product. Vectors: geometric approach B A B′ A′ A vector is represented by a directed segment. • Directed segment is drawn as an arrow. • Different arrows represent the same vector if • they are of the same length and direction. Vectors: geometric approach v B A v B′ A′ −→AB denotes the vector represented by the arrow with tip at B and tail at A. −→AA is called the zero vector and denoted 0. Vectors: geometric approach v B − A v B′ A′ If v = −→AB then −→BA is called the negative vector of v and denoted v. − Vector addition Given vectors a and b, their sum a + b is defined by the rule −→AB + −→BC = −→AC. That is, choose points A, B, C so that −→AB = a and −→BC = b. Then a + b = −→AC. B b a C a b + B′ b A a C ′ a + b A′ The difference of the two vectors is defined as a b = a + ( b). − − b a b − a Properties of vector addition: a + b = b + a (commutative law) (a + b) + c = a + (b + c) (associative law) a + 0 = 0 + a = a a + ( a) = ( a) + a = 0 − − Let −→AB = a. Then a + 0 = −→AB + −→BB = −→AB = a, a + ( a) = −→AB + −→BA = −→AA = 0. − Let −→AB = a, −→BC = b, and −→CD = c. Then (a + b) + c = (−→AB + −→BC) + −→CD = −→AC + −→CD = −→AD, a + (b + c) = −→AB + (−→BC + −→CD) = −→AB + −→BD = −→AD. Parallelogram rule Let −→AB = a, −→BC = b, −−→AB′ = b, and −−→B′C ′ = a. Then a + b = −→AC, b + a = −−→AC ′. -
Unit 6: Matrix Decomposition
Unit 6: Matrix decomposition Juan Luis Melero and Eduardo Eyras October 2018 1 Contents 1 Matrix decomposition3 1.1 Definitions.............................3 2 Singular Value Decomposition6 3 Exercises 13 4 R practical 15 4.1 SVD................................ 15 2 1 Matrix decomposition 1.1 Definitions Matrix decomposition consists in transforming a matrix into a product of other matrices. Matrix diagonalization is a special case of decomposition and is also called diagonal (eigen) decomposition of a matrix. Definition: there is a diagonal decomposition of a square matrix A if we can write A = UDU −1 Where: • D is a diagonal matrix • The diagonal elements of D are the eigenvalues of A • Vector columns of U are the eigenvectors of A. For symmetric matrices there is a special decomposition: Definition: given a symmetric matrix A (i.e. AT = A), there is a unique decomposition of the form A = UDU T Where • U is an orthonormal matrix • Vector columns of U are the unit-norm orthogonal eigenvectors of A • D is a diagonal matrix formed by the eigenvalues of A This special decomposition is known as spectral decomposition. Definition: An orthonormal matrix is a square matrix whose columns and row vectors are orthogonal unit vectors (orthonormal vectors). Orthonormal matrices have the property that their transposed matrix is the inverse matrix. 3 Proposition: if a matrix Q is orthonormal, then QT = Q−1 Proof: consider the row vectors in a matrix Q: 0 1 u1 B . C T u j ::: ju Q = @ . A and Q = 1 n un 0 1 0 1 0 1 u1 hu1; u1i ::: hu1; uni 1 ::: 0 T B . -
Basics of Linear Algebra
Basics of Linear Algebra Jos and Sophia Vectors ● Linear Algebra Definition: A list of numbers with a magnitude and a direction. ○ Magnitude: a = [4,3] |a| =sqrt(4^2+3^2)= 5 ○ Direction: angle vector points ● Computer Science Definition: A list of numbers. ○ Example: Heights = [60, 68, 72, 67] Dot Product of Vectors Formula: a · b = |a| × |b| × cos(θ) ● Definition: Multiplication of two vectors which results in a scalar value ● In the diagram: ○ |a| is the magnitude (length) of vector a ○ |b| is the magnitude of vector b ○ Θ is the angle between a and b Matrix ● Definition: ● Matrix elements: ● a)Matrix is an arrangement of numbers into rows and columns. ● b) A matrix is an m × n array of scalars from a given field F. The individual values in the matrix are called entries. ● Matrix dimensions: the number of rows and columns of the matrix, in that order. Multiplication of Matrices ● The multiplication of two matrices ● Result matrix dimensions ○ Notation: (Row, Column) ○ Columns of the 1st matrix must equal the rows of the 2nd matrix ○ Result matrix is equal to the number of (1, 2, 3) • (7, 9, 11) = 1×7 +2×9 + 3×11 rows in 1st matrix and the number of = 58 columns in the 2nd matrix ○ Ex. 3 x 4 ॱ 5 x 3 ■ Dot product does not work ○ Ex. 5 x 3 ॱ 3 x 4 ■ Dot product does work ■ Result: 5 x 4 Dot Product Application ● Application: Ray tracing program ○ Quickly create an image with lower quality ○ “Refinement rendering pass” occurs ■ Removes the jagged edges ○ Dot product used to calculate ■ Intersection between a ray and a sphere ■ Measure the length to the intersection points ● Application: Forward Propagation ○ Input matrix * weighted matrix = prediction matrix http://immersivemath.com/ila/ch03_dotprodu ct/ch03.html#fig_dp_ray_tracer Projections One important use of dot products is in projections. -
LU Decomposition - Wikipedia, the Free Encyclopedia
LU decomposition - Wikipedia, the free encyclopedia http://en.wikipedia.org/wiki/LU_decomposition From Wikipedia, the free encyclopedia In linear algebra, LU decomposition (also called LU factorization) is a matrix decomposition which writes a matrix as the product of a lower triangular matrix and an upper triangular matrix. The product sometimes includes a permutation matrix as well. This decomposition is used in numerical analysis to solve systems of linear equations or calculate the determinant of a matrix. LU decomposition can be viewed as a matrix form of Gaussian elimination. LU decomposition was introduced by mathematician Alan Turing [1] 1 Definitions 2 Existence and uniqueness 3 Positive definite matrices 4 Explicit formulation 5 Algorithms 5.1 Doolittle algorithm 5.2 Crout and LUP algorithms 5.3 Theoretical complexity 6 Small example 7 Sparse matrix decomposition 8 Applications 8.1 Solving linear equations 8.2 Inverse matrix 8.3 Determinant 9 See also 10 References 11 External links Let A be a square matrix. An LU decomposition is a decomposition of the form where L and U are lower and upper triangular matrices (of the same size), respectively. This means that L has only zeros above the diagonal and U has only zeros below the diagonal. For a matrix, this becomes: An LDU decomposition is a decomposition of the form where D is a diagonal matrix and L and U are unit triangular matrices, meaning that all the entries on the diagonals of L and U LDU decomposition of a Walsh matrix 1 of 7 1/11/2012 5:26 PM LU decomposition - Wikipedia, the free encyclopedia http://en.wikipedia.org/wiki/LU_decomposition are one. -
Arxiv:1105.1185V1 [Math.NA] 5 May 2011 Ento 2.2
ITERATIVE METHODS FOR COMPUTING EIGENVALUES AND EIGENVECTORS MAYSUM PANJU Abstract. We examine some numerical iterative methods for computing the eigenvalues and eigenvectors of real matrices. The five methods examined here range from the simple power iteration method to the more complicated QR iteration method. The derivations, procedure, and advantages of each method are briefly discussed. 1. Introduction Eigenvalues and eigenvectors play an important part in the applications of linear algebra. The naive method of finding the eigenvalues of a matrix involves finding the roots of the characteristic polynomial of the matrix. In industrial sized matrices, however, this method is not feasible, and the eigenvalues must be obtained by other means. Fortunately, there exist several other techniques for finding eigenvalues and eigenvectors of a matrix, some of which fall under the realm of iterative methods. These methods work by repeatedly refining approximations to the eigenvectors or eigenvalues, and can be terminated whenever the approximations reach a suitable degree of accuracy. Iterative methods form the basis of much of modern day eigenvalue computation. In this paper, we outline five such iterative methods, and summarize their derivations, procedures, and advantages. The methods to be examined are the power iteration method, the shifted inverse iteration method, the Rayleigh quotient method, the simultaneous iteration method, and the QR method. This paper is meant to be a survey over existing algorithms for the eigenvalue computation problem. Section 2 of this paper provides a brief review of some of the linear algebra background required to understand the concepts that are discussed. In section 3, the iterative methods are each presented, in order of complexity, and are studied in brief detail. -
Quantum Flatland and Monolayer Graphene from a Viewpoint of Geometric Algebra A
Vol. 124 (2013) ACTA PHYSICA POLONICA A No. 4 Quantum Flatland and Monolayer Graphene from a Viewpoint of Geometric Algebra A. Dargys∗ Center for Physical Sciences and Technology, Semiconductor Physics Institute A. Go²tauto 11, LT-01108 Vilnius, Lithuania (Received May 11, 2013) Quantum mechanical properties of the graphene are, as a rule, treated within the Hilbert space formalism. However a dierent approach is possible using the geometric algebra, where quantum mechanics is done in a real space rather than in the abstract Hilbert space. In this article the geometric algebra is applied to a simple quantum system, a single valley of monolayer graphene, to show the advantages and drawbacks of geometric algebra over the Hilbert space approach. In particular, 3D and 2D Euclidean space algebras Cl3;0 and Cl2;0 are applied to analyze relativistic properties of the graphene. It is shown that only three-dimensional Cl3;0 rather than two-dimensional Cl2;0 algebra is compatible with a relativistic atland. DOI: 10.12693/APhysPolA.124.732 PACS: 73.43.Cd, 81.05.U−, 03.65.Pm 1. Introduction mension has been reduced, one is automatically forced to change the type of Cliord algebra and its irreducible Graphene is a two-dimensional crystal made up of car- representation. bon atoms. The intense interest in graphene is stimulated On the other hand, if the problem was formulated in by its potential application in the construction of novel the Hilbert space terms then, generally, the neglect of nanodevices based on relativistic physics, or at least with one of space dimensions does not spoil the Hilbert space.