Computing Real Square Roots of a Real Matrix* LINEAR ALGEBRA
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
Lecture Notes in One File
Phys 6124 zipped! World Wide Quest to Tame Math Methods Predrag Cvitanovic´ January 2, 2020 Overview The career of a young theoretical physicist consists of treating the harmonic oscillator in ever-increasing levels of abstraction. — Sidney Coleman I am leaving the course notes here, not so much for the notes themselves –they cannot be understood on their own, without the (unrecorded) live lectures– but for the hyperlinks to various source texts you might find useful later on in your research. We change the topics covered year to year, in hope that they reflect better what a graduate student needs to know. This year’s experiment are the two weeks dedicated to data analysis. Let me know whether you would have preferred the more traditional math methods fare, like Bessels and such. If you have taken this course live, you might have noticed a pattern: Every week we start with something obvious that you already know, let mathematics lead us on, and then suddenly end up someplace surprising and highly non-intuitive. And indeed, in the last lecture (that never took place), we turn Coleman on his head, and abandon harmonic potential for the inverted harmonic potential, “spatiotemporal cats” and chaos, arXiv:1912.02940. Contents 1 Linear algebra7 HomeworkHW1 ................................ 7 1.1 Literature ................................ 8 1.2 Matrix-valued functions ......................... 9 1.3 A linear diversion ............................ 12 1.4 Eigenvalues and eigenvectors ...................... 14 1.4.1 Yes, but how do you really do it? . 18 References ................................... 19 exercises 20 2 Eigenvalue problems 23 HomeworkHW2 ................................ 23 2.1 Normal modes .............................. 24 2.2 Stable/unstable manifolds ....................... -
Introduction to Quantum Field Theory
Utrecht Lecture Notes Masters Program Theoretical Physics September 2006 INTRODUCTION TO QUANTUM FIELD THEORY by B. de Wit Institute for Theoretical Physics Utrecht University Contents 1 Introduction 4 2 Path integrals and quantum mechanics 5 3 The classical limit 10 4 Continuous systems 18 5 Field theory 23 6 Correlation functions 36 6.1 Harmonic oscillator correlation functions; operators . 37 6.2 Harmonic oscillator correlation functions; path integrals . 39 6.2.1 Evaluating G0 ............................... 41 6.2.2 The integral over qn ............................ 42 6.2.3 The integrals over q1 and q2 ....................... 43 6.3 Conclusion . 44 7 Euclidean Theory 46 8 Tunneling and instantons 55 8.1 The double-well potential . 57 8.2 The periodic potential . 66 9 Perturbation theory 71 10 More on Feynman diagrams 80 11 Fermionic harmonic oscillator states 87 12 Anticommuting c-numbers 90 13 Phase space with commuting and anticommuting coordinates and quanti- zation 96 14 Path integrals for fermions 108 15 Feynman diagrams for fermions 114 2 16 Regularization and renormalization 117 17 Further reading 127 3 1 Introduction Physical systems that involve an infinite number of degrees of freedom are usually described by some sort of field theory. Almost all systems in nature involve an extremely large number of degrees of freedom. A droplet of water contains of the order of 1026 molecules and while each water molecule can in many applications be described as a point particle, each molecule has itself a complicated structure which reveals itself at molecular length scales. To deal with this large number of degrees of freedom, which for all practical purposes is infinite, we often regard a system as continuous, in spite of the fact that, at certain distance scales, it is discrete. -
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. -
Journal of Computational and Applied Mathematics Exponentials of Skew
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Journal of Computational and Applied Mathematics 233 (2010) 2867–2875 Contents lists available at ScienceDirect Journal of Computational and Applied Mathematics journal homepage: www.elsevier.com/locate/cam Exponentials of skew-symmetric matrices and logarithms of orthogonal matrices João R. Cardoso a,b,∗, F. Silva Leite c,b a Institute of Engineering of Coimbra, Rua Pedro Nunes, 3030-199 Coimbra, Portugal b Institute of Systems and Robotics, University of Coimbra, Pólo II, 3030-290 Coimbra, Portugal c Department of Mathematics, University of Coimbra, 3001-454 Coimbra, Portugal article info a b s t r a c t Article history: Two widely used methods for computing matrix exponentials and matrix logarithms are, Received 11 July 2008 respectively, the scaling and squaring and the inverse scaling and squaring. Both methods become effective when combined with Padé approximation. This paper deals with the MSC: computation of exponentials of skew-symmetric matrices and logarithms of orthogonal 65Fxx matrices. Our main goal is to improve these two methods by exploiting the special structure Keywords: of skew-symmetric and orthogonal matrices. Geometric features of the matrix exponential Skew-symmetric matrix and logarithm and extensions to the special Euclidean group of rigid motions are also Orthogonal matrix addressed. Matrix exponential ' 2009 Elsevier B.V. All rights reserved. Matrix logarithm Scaling and squaring Inverse scaling and squaring Padé approximants 1. Introduction n×n Given a square matrix X 2 R , the exponential of X is given by the absolute convergent power series 1 X X k eX D : kW kD0 n×n X Reciprocally, given a nonsingular matrix A 2 R , any solution of the matrix equation e D A is called a logarithm of A. -
The Exponential Map for the Group of Similarity Transformations and Applications to Motion Interpolation
The exponential map for the group of similarity transformations and applications to motion interpolation Spyridon Leonardos1, Christine Allen–Blanchette2 and Jean Gallier3 Abstract— In this paper, we explore the exponential map The problem of motion interpolation has been studied and its inverse, the logarithm map, for the group SIM(n) of extensively both in the robotics and computer graphics n similarity transformations in R which are the composition communities. Since rotations can be represented by quater- of a rotation, a translation and a uniform scaling. We give a formula for the exponential map and we prove that it is nions, the problem of quaternion interpolation has been surjective. We give an explicit formula for the case of n = 3 investigated, an approach initiated by Shoemake [21], [22], and show how to efficiently compute the logarithmic map. As who extended the de Casteljau algorithm to the 3-sphere. an application, we use these algorithms to perform motion Related work was done by Barr et al. [2]. Moreover, Kim interpolation. Given a sequence of similarity transformations, et al. [13], [14] corrected bugs in Shoemake and introduced we compute a sequence of logarithms, then fit a cubic spline 4 that interpolates the logarithms and finally, we compute the various kinds of splines on the 3-sphere in R , using the interpolating curve in SIM(3). exponential map. Motion interpolation and rational motions have been investigated by Juttler¨ [9], [10], Juttler¨ and Wagner I. INTRODUCTION [11], [12], Horsch and Juttler¨ [8], and Roschel¨ [20]. Park and Ravani [18], [19] also investigated Bezier´ curves on The exponential map is ubiquitous in engineering appli- Riemannian manifolds and Lie groups and in particular, the cations ranging from dynamical systems and robotics to group of rotations. -
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 . -
Linear Algebra Review
Appendix A Linear Algebra Review In this appendix, we review results from linear algebra that are used in the text. The results quoted here are mostly standard, and the proofs are mostly omitted. For more information, the reader is encouraged to consult such standard linear algebra textbooks as [HK]or[Axl]. Throughout this appendix, we let Mn.C/ denote the space of n n matrices with entries in C: A.1 Eigenvectors and Eigenvalues n For any A 2 Mn.C/; a nonzero vector v in C is called an eigenvector for A if there is some complex number such that Av D v: An eigenvalue for A is a complex number for which there exists a nonzero v 2 Cn with Av D v: Thus, is an eigenvalue for A if the equation Av D v or, equivalently, the equation .A I /v D 0; has a nonzero solution v: This happens precisely when A I fails to be invertible, which is precisely when det.A I / D 0: For any A 2 Mn.C/; the characteristic polynomial p of A is given by A previous version of this book was inadvertently published without the middle initial of the author’s name as “Brian Hall”. For this reason an erratum has been published, correcting the mistake in the previous version and showing the correct name as Brian C. Hall (see DOI http:// dx.doi.org/10.1007/978-3-319-13467-3_14). The version readers currently see is the corrected version. The Publisher would like to apologize for the earlier mistake. -
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. -
Arxiv:2004.07192V2 [Quant-Ph] 14 May 2021
Two-way covert quantum communication in the microwave regime R. Di Candia,1, ∗ H. Yi˘gitler,1 G. S. Paraoanu,2 and R. J¨antti1 1Department of Communications and Networking, Aalto University, Espoo, 02150 Finland 2QTF Centre of Excellence, Department of Applied Physics, Aalto University School of Science, FI-00076 AALTO, Finland Quantum communication addresses the problem of exchanging information across macroscopic distances by employing encryption techniques based on quantum mechanical laws. Here, we ad- vance a new paradigm for secure quantum communication by combining backscattering concepts with covert communication in the microwave regime. Our protocol allows communication between Alice, who uses only discrete phase modulations, and Bob, who has access to cryogenic microwave technology. Using notions of quantum channel discrimination and quantum metrology, we find the ultimate bounds for the receiver performance, proving that quantum correlations can enhance the signal-to-noise ratio by up to 6 dB. These bounds rule out any quantum illumination advantage when the source is strongly amplified, and shows that a relevant gain is possible only in the low photon-number regime. We show how the protocol can be used for covert communication, where the carrier signal is indistinguishable from the thermal noise in the environment. We complement our information-theoretic results with a feasible experimental proposal in a circuit QED platform. This work makes a decisive step toward implementing secure quantum communication concepts in the previously uncharted 1 − 10 GHz frequency range, in the scenario when the disposable power of one party is severely constrained. I. INTRODUCTION range. For instance, QKD security proofs require level of noises which at room temperature are reachable only by frequencies at least in the Terahertz band [8]. -
Randomized Linear Algebra Approaches to Estimate the Von Neumann Entropy of Density Matrices
Randomized Linear Algebra Approaches to Estimate the Von Neumann Entropy of Density Matrices Eugenia-Maria Kontopoulou Ananth Grama Wojciech Szpankowski Petros Drineas Purdue University Purdue University Purdue University Purdue University Computer Science Computer Science Computer Science Computer Science West Lafayette, IN, USA West Lafayette, IN, USA West Lafayette, IN, USA West Lafayette, IN, USA Email: [email protected] Email: [email protected] Email: [email protected] Email: [email protected] Abstract—The von Neumann entropy, named after John von Motivated by the above discussion, we seek numerical Neumann, is the extension of classical entropy concepts to the algorithms that approximate the von Neumann entropy of large field of quantum mechanics and, from a numerical perspective, density matrices, e.g., symmetric positive definite matrices can be computed simply by computing all the eigenvalues of a 3 density matrix, an operation that could be prohibitively expensive with unit trace, faster than the trivial O(n ) approach. Our for large-scale density matrices. We present and analyze two algorithms build upon recent developments in the field of randomized algorithms to approximate the von Neumann entropy Randomized Numerical Linear Algebra (RandNLA), an in- of density matrices: our algorithms leverage recent developments terdisciplinary research area that exploits randomization as a in the Randomized Numerical Linear Algebra (RandNLA) liter- computational resource to develop improved algorithms for ature, such as randomized trace -
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.