Skew-Symmetric Endomorphisms in M1,3: a Unified Canonical Form with Applications to Conformal Geometry
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Solution to Homework 5
Solution to Homework 5 Sec. 5.4 1 0 2. (e) No. Note that A = is in W , but 0 2 0 1 1 0 0 2 T (A) = = 1 0 0 2 1 0 is not in W . Hence, W is not a T -invariant subspace of V . 3. Note that T : V ! V is a linear operator on V . To check that W is a T -invariant subspace of V , we need to know if T (w) 2 W for any w 2 W . (a) Since we have T (0) = 0 2 f0g and T (v) 2 V; so both of f0g and V to be T -invariant subspaces of V . (b) Note that 0 2 N(T ). For any u 2 N(T ), we have T (u) = 0 2 N(T ): Hence, N(T ) is a T -invariant subspace of V . For any v 2 R(T ), as R(T ) ⊂ V , we have v 2 V . So, by definition, T (v) 2 R(T ): Hence, R(T ) is also a T -invariant subspace of V . (c) Note that for any v 2 Eλ, λv is a scalar multiple of v, so λv 2 Eλ as Eλ is a subspace. So we have T (v) = λv 2 Eλ: Hence, Eλ is a T -invariant subspace of V . 4. For any w in W , we know that T (w) is in W as W is a T -invariant subspace of V . Then, by induction, we know that T k(w) is also in W for any k. k Suppose g(T ) = akT + ··· + a1T + a0, we have k g(T )(w) = akT (w) + ··· + a1T (w) + a0(w) 2 W because it is just a linear combination of elements in W . -
Refinements of the Weyl Tensor Classification in Five Dimensions
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Ghent University Academic Bibliography Home Search Collections Journals About Contact us My IOPscience Refinements of the Weyl tensor classification in five dimensions This article has been downloaded from IOPscience. Please scroll down to see the full text article. 2012 Class. Quantum Grav. 29 155016 (http://iopscience.iop.org/0264-9381/29/15/155016) View the table of contents for this issue, or go to the journal homepage for more Download details: IP Address: 157.193.53.227 The article was downloaded on 16/07/2012 at 16:13 Please note that terms and conditions apply. IOP PUBLISHING CLASSICAL AND QUANTUM GRAVITY Class. Quantum Grav. 29 (2012) 155016 (50pp) doi:10.1088/0264-9381/29/15/155016 Refinements of the Weyl tensor classification in five dimensions Alan Coley1, Sigbjørn Hervik2, Marcello Ortaggio3 and Lode Wylleman2,4,5 1 Department of Mathematics and Statistics, Dalhousie University, Halifax, Nova Scotia B3H 3J5, Canada 2 Faculty of Science and Technology, University of Stavanger, N-4036 Stavanger, Norway 3 Institute of Mathematics, Academy of Sciences of the Czech Republic, Zitnˇ a´ 25, 115 67 Prague 1, Czech Republic 4 Department of Mathematical Analysis EA16, Ghent University, 9000 Gent, Belgium 5 Department of Mathematics, Utrecht University, 3584 CD Utrecht, The Netherlands E-mail: [email protected], [email protected], [email protected] and [email protected] Received 31 March 2012, in final form 21 June 2012 Published 16 July 2012 Online at stacks.iop.org/CQG/29/155016 Abstract We refine the null alignment classification of the Weyl tensor of a five- dimensional spacetime. -
Kinematics of Visually-Guided Eye Movements
Kinematics of Visually-Guided Eye Movements Bernhard J. M. Hess*, Jakob S. Thomassen Department of Neurology, University Hospital Zurich, Zurich, Switzerland Abstract One of the hallmarks of an eye movement that follows Listing’s law is the half-angle rule that says that the angular velocity of the eye tilts by half the angle of eccentricity of the line of sight relative to primary eye position. Since all visually-guided eye movements in the regime of far viewing follow Listing’s law (with the head still and upright), the question about its origin is of considerable importance. Here, we provide theoretical and experimental evidence that Listing’s law results from a unique motor strategy that allows minimizing ocular torsion while smoothly tracking objects of interest along any path in visual space. The strategy consists in compounding conventional ocular rotations in meridian planes, that is in horizontal, vertical and oblique directions (which are all torsion-free) with small linear displacements of the eye in the frontal plane. Such compound rotation-displacements of the eye can explain the kinematic paradox that the fixation point may rotate in one plane while the eye rotates in other planes. Its unique signature is the half-angle law in the position domain, which means that the rotation plane of the eye tilts by half-the angle of gaze eccentricity. We show that this law does not readily generalize to the velocity domain of visually-guided eye movements because the angular eye velocity is the sum of two terms, one associated with rotations in meridian planes and one associated with displacements of the eye in the frontal plane. -
Spectral Coupling for Hermitian Matrices
Spectral coupling for Hermitian matrices Franc¸oise Chatelin and M. Monserrat Rincon-Camacho Technical Report TR/PA/16/240 Publications of the Parallel Algorithms Team http://www.cerfacs.fr/algor/publications/ SPECTRAL COUPLING FOR HERMITIAN MATRICES FRANC¸OISE CHATELIN (1);(2) AND M. MONSERRAT RINCON-CAMACHO (1) Cerfacs Tech. Rep. TR/PA/16/240 Abstract. The report presents the information processing that can be performed by a general hermitian matrix when two of its distinct eigenvalues are coupled, such as λ < λ0, instead of λ+λ0 considering only one eigenvalue as traditional spectral theory does. Setting a = 2 = 0 and λ0 λ 6 e = 2− > 0, the information is delivered in geometric form, both metric and trigonometric, associated with various right-angled triangles exhibiting optimality properties quantified as ratios or product of a and e. The potential optimisation has a triple nature which offers two j j e possibilities: in the case λλ0 > 0 they are characterised by a and a e and in the case λλ0 < 0 a j j j j by j j and a e. This nature is revealed by a key generalisation to indefinite matrices over R or e j j C of Gustafson's operator trigonometry. Keywords: Spectral coupling, indefinite symmetric or hermitian matrix, spectral plane, invariant plane, catchvector, antieigenvector, midvector, local optimisation, Euler equation, balance equation, torus in 3D, angle between complex lines. 1. Spectral coupling 1.1. Introduction. In the work we present below, we focus our attention on the coupling of any two distinct real eigenvalues λ < λ0 of a general hermitian or symmetric matrix A, a coupling called spectral coupling. -
Section 18.1-2. in the Next 2-3 Lectures We Will Have a Lightning Introduction to Representations of finite Groups
Section 18.1-2. In the next 2-3 lectures we will have a lightning introduction to representations of finite groups. For any vector space V over a field F , denote by L(V ) the algebra of all linear operators on V , and by GL(V ) the group of invertible linear operators. Note that if dim(V ) = n, then L(V ) = Matn(F ) is isomorphic to the algebra of n×n matrices over F , and GL(V ) = GLn(F ) to its multiplicative group. Definition 1. A linear representation of a set X in a vector space V is a map φ: X ! L(V ), where L(V ) is the set of linear operators on V , the space V is called the space of the rep- resentation. We will often denote the representation of X by (V; φ) or simply by V if the homomorphism φ is clear from the context. If X has any additional structures, we require that the map φ is a homomorphism. For example, a linear representation of a group G is a homomorphism φ: G ! GL(V ). Definition 2. A morphism of representations φ: X ! L(V ) and : X ! L(U) is a linear map T : V ! U, such that (x) ◦ T = T ◦ φ(x) for all x 2 X. In other words, T makes the following diagram commutative φ(x) V / V T T (x) U / U An invertible morphism of two representation is called an isomorphism, and two representa- tions are called isomorphic (or equivalent) if there exists an isomorphism between them. Example. (1) A representation of a one-element set in a vector space V is simply a linear operator on V . -
Local Stability
Chapter 4 Local stability It does not say in the Bible that all laws of nature are ex- pressible linearly. — Enrico Fermi (R. Mainieri and P. Cvitanovic)´ o far we have concentrated on describing the trajectory of a single initial point. Our next task is to define and determine the size of a neighborhood S of x(t). We shall do this by assuming that the flow is locally smooth and by describing the local geometry of the neighborhood by studying the flow linearized around x(t). Nearby points aligned along the stable (contracting) directions remain t in the neighborhood of the trajectory x(t) = f (x0); the ones to keep an eye on are the points which leave the neighborhood along the unstable directions. As we shall demonstrate in chapter 21, the expanding directions matter in hyperbolic systems. The repercussions are far-reaching. As long as the number of unstable directions is finite, the same theory applies to finite-dimensional ODEs, state space volume preserving Hamiltonian flows, and dissipative, volume contracting infinite-dim- ensional PDEs. In order to streamline the exposition, in this chapter all examples are collected in sect. 4.8. We strongly recommend that you work through these examples: you can get to them and back to the text by clicking on the [example] links, such as example 4.8 p. 96 4.1 Flows transport neighborhoods As a swarm of representative points moves along, it carries along and distorts neighborhoods. The deformation of an infinitesimal neighborhood is best un- derstood by considering a trajectory originating near x0 = x(0), with an initial 82 CHAPTER 4. -
A Unified Canonical Form with Applications to Conformal Geometry
Skew-symmetric endomorphisms in M1,3: A unified canonical form with applications to conformal geometry Marc Mars and Carlos Pe´on-Nieto Instituto de F´ısica Fundamental y Matem´aticas, Universidad de Salamanca Plaza de la Merced s/n 37008, Salamanca, Spain Abstract We derive a canonical form for skew-symmetric endomorphisms F in Lorentzian vector spaces of dimension three and four which covers all non-trivial cases at once. We analyze its invariance group, as well as the connection of this canonical form with duality rotations of two-forms. After reviewing the relation between these endomorphisms and the algebra of 2 2 2 conformal Killing vectors of S , CKill S , we are able to also give a canonical form for an arbitrary element ξ ∈ CKill S along with its invariance group. The construction allows us to obtain explicitly the change of basis that transforms any given F into its canonical form. For any non-trivial ξ we construct, via its canonical form, adapted coordinates that allow us to study its properties in depth. Two applications are worked out: we determine explicitly for which metrics, among a natural class of spaces of constant curvature, a given ξ is a Killing vector and solve all local TT (traceless and transverse) tensors that satisfy the Killing Initial Data equation for ξ. In addition to their own interest, the present results will be a basic ingredient for a subsequent generalization to arbitrary dimensions. 1 Introduction Finding a canonical form for the elements of a certain set is often an interesting problem to solve, since it is a powerful tool for both computations and mathematical analysis. -
Adiabatic Invariants of the Linear Hamiltonian Systems with Periodic Coefficients
JOURNAL OF DIFFERENTIAL EQUATIONS 42, 47-71 (1981) Adiabatic Invariants of the Linear Hamiltonian Systems with Periodic Coefficients MARK LEVI Department of Mathematics, Duke University, Durham, North Carolina 27706 Received November 19, 1980 0. HISTORICAL REMARKS In 1911 Lorentz and Einstein had offered an explanation of the fact that the ratio of energy to the frequency of radiation of an atom remains constant. During the long time intervals separating two quantum jumps an atom is exposed to varying surrounding electromagnetic fields which should supposedly change the ratio. Their explanation was based on the fact that the surrounding field varies extremely slowly with respect to the frequency of oscillations of the atom. The idea can be illstrated by a slightly simpler model-a pendulum (instead of Bohr’s model of an atom) with a slowly changing length (slowly with respect to the frequency of oscillations of the pendulum). As it turns out, the ratio of energy of the pendulum to its frequency changes very little if its length varies slowly enough from one constant value to another; the pendulum “remembers” the ratio, and the slower the change the better the memory. I had learned this interesting historical remark from Wasow [15]. Those parameters of a system which remain nearly constant during a slow change of a system were named by the physicists the adiabatic invariants, the word “adiabatic” indicating that one parameter changes slowly with respect to another-e.g., the length of the pendulum with respect to the phaze of oscillations. Precise definition of the concept will be given in Section 1. -
A Criterion for the Existence of Common Invariant Subspaces Of
Linear Algebra and its Applications 322 (2001) 51–59 www.elsevier.com/locate/laa A criterion for the existence of common invariant subspaces of matricesୋ Michael Tsatsomeros∗ Department of Mathematics and Statistics, University of Regina, Regina, Saskatchewan, Canada S4S 0A2 Received 4 May 1999; accepted 22 June 2000 Submitted by R.A. Brualdi Abstract It is shown that square matrices A and B have a common invariant subspace W of di- mension k > 1 if and only if for some scalar s, A + sI and B + sI are invertible and their kth compounds have a common eigenvector, which is a Grassmann representative for W.The applicability of this criterion and its ability to yield a basis for the common invariant subspace are investigated. © 2001 Elsevier Science Inc. All rights reserved. AMS classification: 15A75; 47A15 Keywords: Invariant subspace; Compound matrix; Decomposable vector; Grassmann space 1. Introduction The lattice of invariant subspaces of a matrix is a well-studied concept with sev- eral applications. In particular, there is substantial information known about matrices having the same invariant subspaces, or having chains of common invariant subspac- es (see [7]). The question of existence of a non-trivial invariant subspace common to two or more matrices is also of interest. It arises, among other places, in the study of irreducibility of decomposable positive linear maps on operator algebras [3]. It also comes up in what might be called Burnside’s theorem, namely that if two square ୋ Research partially supported by NSERC research grant. ∗ Tel.: +306-585-4771; fax: +306-585-4020. E-mail address: [email protected] (M. -
The Semi-Simplicity Manifold of Arbitrary Operators
THE SEMI-SIMPLICITY MANIFOLD OF ARBITRARY OPERATORS BY SHMUEL KANTOROVITZ(i) Introduction. In finite dimensional complex Euclidian space, any linear operator has a unique maximal invariant subspace on which it is semi-simple. It is easily described by means of the Jordan decomposition theorem. The main result of this paper (Theorem 2.1) is a generalization of this fact to infinite di- mensional reflexive Banach space, for arbitrary bounded operators T with real spectrum. Our description of the "semi-simplicity manifold" for T is entirely analytic and basis-free, and seems therefore to be a quite natural candidate for such generalizations to infinite dimension. A similar generalization to infinite dimension of the concepts of Jordan cells and Weyr characteristic will be presented elsewhere. The theory is motivated and illustrated by examples in §3. Notations. The following notations are fixed throughout this paper, and will be used without further explanation. R: the field of real numbers. C : the field of complex numbers. âd : the sigma-algebra of all Borel subsets of R. C(R) : the space of all continuous complex valued functions on R. L"(R) : the usual Lebesgue spaces on R (p = 1). j : integration over R. /: the Fourier transform of / e L1(R). X : an arbitrary complex Banach space. X*: the conjugate of X. B(X) : the Banach algebra of all bounded linear operators acting on X. I : the identity operator on X. | • | : the original norms on X, X* and B(X) (new norms, to be defined, will be denoted by double bars). | • \y, | • |œ: the L\R) andL™^) norms (respectively). -
The Common Invariant Subspace Problem
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 384 (2004) 1–7 www.elsevier.com/locate/laa The common invariant subspace problem: an approach via Gröbner bases Donu Arapura a, Chris Peterson b,∗ aDepartment of Mathematics, Purdue University, West Lafayette, IN 47907, USA bDepartment of Mathematics, Colorado State University, Fort Collins, CO 80523, USA Received 24 March 2003; accepted 17 November 2003 Submitted by R.A. Brualdi Abstract Let A be an n × n matrix. It is a relatively simple process to construct a homogeneous ideal (generated by quadrics) whose associated projective variety parametrizes the one-dimensional invariant subspaces of A. Given a finite collection of n × n matrices, one can similarly con- struct a homogeneous ideal (again generated by quadrics) whose associated projective variety parametrizes the one-dimensional subspaces which are invariant subspaces for every member of the collection. Gröbner basis techniques then provide a finite, rational algorithm to deter- mine how many points are on this variety. In other words, a finite, rational algorithm is given to determine both the existence and quantity of common one-dimensional invariant subspaces to a set of matrices. This is then extended, for each d, to an algorithm to determine both the existence and quantity of common d-dimensional invariant subspaces to a set of matrices. © 2004 Published by Elsevier Inc. Keywords: Eigenvector; Invariant subspace; Grassmann variety; Gröbner basis; Algorithm 1. Introduction Let A and B be two n × n matrices. In 1984, Dan Shemesh gave a very nice procedure for constructing the maximal A, B invariant subspace, N,onwhichA and B commute. -
Some Open Problems in the Theory of Subnormal Operators
Holomorphic Spaces MSRI Publications Volume 33, 1998 Some Open Problems in the Theory of Subnormal Operators JOHN B. CONWAY AND LIMING YANG Abstract. Subnormal operators arise naturally in complex function the- ory, differential geometry, potential theory, and approximation theory, and their study has rich applications in many areas of applied sciences as well as in pure mathematics. We discuss here some research problems concern- ing the structure of such operators: subnormal operators with finite-rank self-commutator, connections with quadrature domains, invariant subspace structure, and some approximation problems related to the theory. We also present some possible approaches for the solution of these problems. Introduction A bounded linear operator S on a separable Hilbert space H is called subnor- mal if there exists a normal operator N on a Hilbert space K containing H such that NH ⊂ H and NjH = S. The operator S is called cyclic if there exists an x in H such that H = closfp(S)x : p is a polynomialg; and is called rationally cyclic if there exists an x such that H = closfr(S)x : r is a rational function with poles off σ(S)g: The operator S is pure if S has no normal summand and is irreducible if S is not unitarily equivalent to a direct sum of two nonzero operators. The theory of subnormal operators provides rich applications in many areas, since many natural operators that arise in complex function theory, differential geometry, potential theory, and approximation theory are subnormal operators. Many deep results have been obtained since Halmos introduced the concept of a subnormal operator.