What is complete integrability in quantum mechanics

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "What is complete integrability in quantum mechanics"

Transcription

1 INTERNATIONAL SOLVAY INSTITUTES FOR PHYSICS AND CHEMISTRY Proceedings of the Symposium Henri Poincaré (Brussels, 8-9 October 2004) What is complete integrability in quantum mechanics L. D. Faddeev St. Petersburg Department of Steklov, Mathematical Institute, Russian Academy of Sciences

2 When I was invited to speak on Solvay Conference, dedicated to the 150 anniversary of H. Poincare, I faced a noneasy problem. Being mathematical physicist, dealing with problems, stemming from quantum theory, I could not find a suitable subject to speak about. Finally I decided to discuss the notion of complete integrability, central to great works of H. Poincare, in its relation to quantum mechanics. It happened, that in the last 30 years interest to complete integrability was revived due to the development of the Inverse Scattering Method in the theory of solitons. It was realized, that many soliton equations have Hamiltonian formulation and are completely integrable see e. g. [1] and references there. The quantization of these dynamical models led to beautiful mathematical structures as Yang-Baxter relation and was the base for the formulation of Quantum Groups. Thus it became natural to understand what integrability means in quantum domain. Based on this development (see e. g. [2] and references there) I present here some general considerations and illustrate them by the fundamental example of quantum soliton theory magnetic chain, following the exposition in [2], where one can find more details. Thus though this text is not very original, I hope that it will stimulate attentive reader to think about the exact nature of the complete integrability in quantum theory. The notion of complete integrability assumes central place in classical mechanics. Almost all great names in mechanics are associated with development of this notion and it is vividly discussed in textbooks. It is not so in quantum mechanics. In this essay I shall discuss several reasons for this phenomenon and after that present an instructive example of passage from classical mechanical treatment of a particular completely integrable dynamical model to quantum description retaining the integrability intact. The relation between quantum and classical pictures are that of deformation and contraction, Planck constant playing the role of the deformation parameter (see e. g. [3]). The construction of a quantum variant of a given dynamical model, known in classical formulation (quantization) is not unique. I use to say, that the quantization is rather an art than science. Thus the transfer of the features of classical mechanics into their quantum analogue is not automatic. Moreover, these features could be modified in this passage. Here we shall illustrate this by the notion of complete integrability. For that we should first remind the basic aspects of the formulation of mechanical system. The main objects in this formulation are the algebra of observables and the evolution in time. In classical mechanics we begin with phase space Γ, dim Γ = 2n, endowed by a nondegenerate 2-form ω = 2n i,k Ω ik d ξ i d ξ k. Here ξ i, i = 1,... 2n are (local) coordinates on Γ. Real functions f : Γ R constitute the algebra of observables A 0. Any given observable g defines one parameter family of automorphisms of A 0 by means of the equation for an arbitrary observable f df(s) ds = {g, f(s)}, f(0) = f, where the Poisson bracket {, } is constructed by means of matrix Ω ik inverse 2

3 to Ω ik : ik f g {f, g} = Ω ξ i ξ k The Poisson bracket introduces the structure of a Lie algebra into the algebra of observables. A particular observable H, called Hamiltonian, defines in this way the time evolution. The equation df = {H, f} (1) dt is called the equation of motion. In conventional formulation of quantum mechanics algebra of observables A h is that of linear selfadjoint operators A, B,... in a complex Hilbert space H. The role of Lie operation plays the commutator [A, B] = i (AB BA). Here i = 1 and real parameter is called Planck constant. The automorphisms are given by one parameter families of unitary operators. In particular Hamiltonian H defines the evolution operator U(t) = e iht/ so that the solution of the equation of motion is given by d dt A(t) = i [HA(t) A(t)H] h A(t) = U 1 (t)a(0)u(t). One cannot help observing, that this explicit formula is much nicer, than the description of the solution of equation (1) in terms of the generating function. We can now turn to the discussion of complete integrability. I shall remind two conventional definitions of this notion in terms of classical mechanics and then observe if they have a natural quantum counterpart. We shall see, that the direct (and may be too naive) reformulation will not be sensible. So let the classical mechanical system (Γ 2n, ω, H) be given. The first definition of complete integrability requires the existence of n 1 functionally independent observables Q i, i = 1,..., n 1 which Poisson commute with H and among themselves {H, Q i } = 0, {Q i, Q k } = 0. Observables Q i are called commuting conservation laws. In the second more elaborate definition one requires the existence of change of variables V such that V (ξ) = (α, I), where angle-action variables α i, I i, i = 1,... n constitute two commuting subsets and Hamiltonian H becomes function of actions I only H = H(I). Both these definitions almost immediately fail if we try to formulate them in quantum terms. For the first one it is just not clear what is the definition of the number of degrees of freedom n. Indeed all infinite dimensional separable 3

4 Hilbert spaces are unitary equivalent, so there is no place for the quantum definition of number of degrees of freedom if we stick to the definition of algebra of observables given above. In the case of the second definition the difficulty is more subtle. It turns out that nonintegrable classical systems could become integrable if we naively generalize the second definition of integrability. Indeed, in the theory of scattering of a particle by potential we deal with the phase space Γ = R 6 with canonical variables momenta and coordinates p 1, p 2, p 3, q 1, q 2, q 3 (we treat particle in the 3-dimensional space). The Hamiltonian H(p, q) = p2 2m + v(q) contains a potential function v(q), which is supposed to vanish rapidly enough when q. In general case when v(q) does not satisfy any symmetry (like spherical symmetry v(q) = v( q )), the classical system is not integrable. On the other hand the quantum mechanical Hamiltonian H = 2 2m + v(x) in L 2 (R 3 ) is unitarily equivalent to the operator H 0, defined in L 2 (R 3 ) C N as H 0 ψ(p) = p2 2m ψ(p), H 0 φ n = κ 2 nφ n, which is a function of action variable momentum p of a free particle and energies κ 2 n of bound states. This equivalence is established by the wave operator of quantum scattering theory. Thus either we are to admit, that quantum mechanics is more regular than the classical mechanics, or restrict the admissible automorphisms so that the wave operator will be excluded. I think, that after this general discussion it become clear, that the definition of the algebra of observables of quantum system given above is not detailed enough. And of course we know, that in dealing with concrete quantum systems we begin with the set of distinguished observables momenta and coordinate operators of a particle, spin operators for magnets etc. Thus the Hilbert space and its topology is not the first basic object. Rather it appears as a carrier of representation for these distinguished variables. For instance the linear phase space Γ 2n leads after quantization to the Hilbert space H = L 2 (R n ), where the Lie algebra of linear symplectic transformations is represented, with explicit indication of the number of degrees of freedom. After this representation is chosen we are to define algebra of observables, generated by basic variables, and admissible automorphisms of this algebra. In particular such definition is to exclude the wave operator. I do not know such definition yet. Instead I shall present an instructive example of a classical integrable model, for which the integrability is retained in a natural sense. Such examples will be indispensable for the future definition of the quantum integrability. The model is called spin chain. The phase space Γ is given by Γ = S 2 S 2... S } {{ } 2. N 4

5 Each 2-sphere S 2 corresponds to a dynamical variable called spin and product means, that we deal with N spins distributed along the discrete circle Z N, n + N n. The Poisson structure on Γ is defined in local way: the variables s i n : i = 1, 2, 3, n = 1,... N with constraint (s 1 n) 2 + (s 2 n) 2 + (s 3 n) 2 = s 2 corresponding to different spins commute {s i n, s k m} = 0, n m. The bracket for a given spin n is given by where ɛ ikj is antisymmetric tensor. It is clear, that {s i n, s k n} = ɛ ikj s j n, (2) dim Γ N = 2N, so we deal with the system of N degrees of freedom. Quantization of bracket (2) is given by the representation of the Lie algebra SU(2). These representations D s are labeled by semiinteger and have dimensions s = 0, 1 2, 1,... dim D s = 2s + 1. The variables s i n become matrices with commutation relations [s i n, s k n] = iɛ ikj s j n (3) (we do not write explicitly using units = 1). For instance in the case s = 1/2 the Hilbert space is C 2 and spin variables are given by s i n = 1 2 σ i, where σ 1 = ( ) 0 1, σ = ( ) ( ) 0 i 1 0, σ i 0 3 =. 0 1 The full Hilbert space of the model is given by the tensor product H = D s1 D s2... D sn. In what follows we shall consider the isotropic case where all labels s 1, s 2,... s N are equal. Thus the role of distinguished observables is played by generators of the algebra SU(2) for each component of the phase space. Now we should introduce a relevant Hamiltonian. According to the general technique of the Inverse Scattering Method (see [2]) it is done as follows. Consider an auxiliary linear problem ψ n+1 = L n (λ)ψ n, (4) 5

6 where ( ) ( ) ψ 1 ψ n = n λ + is 3 ψn 2, L n = n is n is + n λ is 3, (5) n λ complex ( spectral ) parameter and s ± n = s 1 n ± is 2 n. The linear system (4) has monodromy around our circle T (λ) = L N (λ)l N 1 (λ)... L 1 (λ). Now the following fundamental property takes place: the invariant F (λ) = tr T (λ) defines a commuting family of observables {F (λ), F (µ)} = 0. (6) We shall not proof this here and just refer to [1]. However more instructive proof of similar property in quantum case will be given below. It is clear from construction, that F (λ) is a polinom in λ N 1 F (λ) = 2λ N + λ k Q k+1 so the observables Q k, k = 1,... N define functionally independent commuting set {Q i, Q k } = 0, i, k = 1,... N. Each of them can be taken as a Hamiltonian of a completely integrable system. However there is another possibility to choose a more beautiful Hamiltonian. Observe, that matrix L n is degenerate when λ 2 + s 2 = 0. Thus L n (is) has rank 1 and so it is easy to calculate the monodromy. In particular we get k=0 ln F (is) 2 = n ln(s 2 + i s i ns i n+1). The RHS is a sum of terms describing the near neighbour interaction along our chain. The Hamiltonian H = n ln(s 2 + s i ns i n+1) (7) defines the classical completely integrable dynamical system, which we already called spin chain. The construction of the quantum counterpart is based on the fact, that there is no problem in quantization of the matrix L k (λ) indeed, it contains the dynamical variables s n linearly and we do not face the problem of factor ordering. We shall prove, that the quantum analogue of the monodromy M(λ), defined exactly by some formula, as the classical one (5), also generate a family of commuting operators. For this we begin by observation that the set of commutation 6

7 relations for the matrix elements of matrix operator L n (x) can be written in an elegant way. Consider L n (x) as a matrix in tensor product D s C 2. We shall call the first factor D s the quantum space and the second C 2 the auxiliary space. Let R(λ) be a matrix in the product of two auxiliary spaces C 2 C 2 R(λ) = λi + ip. (8) Here P is a permutation P (a b) = b a. Then the following relation holds in the space C 2 C 2 D s where R 12 (λ µ)l 1 n(λ)l 2 n(µ) = L 2 n(µ)l 1 n(λ)r 12 (λ µ), (9) R 12 (λ µ) = R(λ µ) I, L 2 n(λ) = I L n (λ) and L 1 n acts nontrivially in the product of the first auxiliary space and D s. The relation (9) can be checked easily by means of the basic relation (3). Now taking into account, that matrix elements of L 1 n(λ) and L 2 m(µ) for n m commute, we find that the monodromy M(λ) = L N (λ)... L 1 (λ) (10) considered as a matrix in H C 2 satisfy the same set of relations as a local factor L n (λ) R 12 (λ µ)m 1 (λ)m 2 (µ) = M 2 (µ)m 1 (λ)r 12 (λ µ) from which it follows immediately, that F (λ) = tr M(λ), where trace is taken over the auxiliary space, defines commuting set of operators [F (λ), F (µ)] = 0. In fact the derivation of the classical commutativity (6) can be derived by the contraction 0. Thus we have the set of N commuting operators Q j, j = 1,... N, being coefficients of the polinom F (λ). But as in the classical case we can find a more beautiful Hamiltonian which is a sum of local terms. Begin with the case s = 1/2, so that D s = C 2 and quantum and auxiliary spaces coincide. It follows from the definition, that L( i 2 ) = ip, where P is permutation matrix. Using this one can show, that F ( i 2 ) is a shift operator in H and d dλ F (λ)f 1 (λ) λ=i/2 = H 1/2 n,n+1, 7

8 where Thus the Hamiltonian H 1/2 n,n+1 = 3 s i ns i n I. i=1 H = n H 1/2 n,n+1 is a sum of terms, describing the interaction of neighbouring spins. This Hamiltonian is well known in the theory of magnets and corresponding dynamical system is called Heisenberg spin chain. The case of higher spins requires more work. We begin by the statement, that the basic relation (9) is a particular case of a more general one R s1,s2 (λ µ)r s1,s3 (λ σ)r s2,s2 (µ σ) = taking place in the space = R s2,s2 (µ σ)r s1,s3 (λ σ)r s1,s2 (λ µ), (11) D 123 = D s1 D s2 D s3, where each factor R s,s acts nontrivially in two of D s, corresponding to its spin labels. The relation (9) corresponds to the case s 1 = 1/2, s 2 = 1/2, s 3 = 1/2, where R 1/2,1/2 is given by (8) and L n (λ) = R 1/2,1/2 (λ i/2) in case of s = 1/2. The formula (11) realizes the concrete representation of the abstract Yang-Baxter relation R 12 (λ µ)r 13 (λ σ)r 23 (µ σ) = R 23 (µ σr 13 (λ σ)r 12 (λ µ), which plays the fundamental role in the theory of quantum integrable models and quantum groups. Using the relation (11) for s 1 = 1/2, s 2 = s, s 3 = s and taking into account, that we know R 1/2,s we get the linear equation for R s,s (λ). The solution can be written as R s,s (λ) = ip r(j, λ). Here operator J is defined by the equation ( s 1 + s 2 ) 2 = J(J + 1), (12) where s 1 and s 2 are spin vectors in the first and second D s. The explicit formula for r looks as follows Γ(J iλ) r(j, λ) = Γ(J + 1 iλ). Now the local Hamiltonian can be constructed due to the relation R s,s (0) = ip as Hn,n+1 s = i d dλ ln r(j n,n+1, λ) λ=0 = 2ψ(J n,n+1 + 1), where J n,n+1 is constructed by formula analogous to (12) with s 1 = s n and s 2 = s n+1. Taking into account, that J n,n+1 (J n,n+1 + 1) = 2( s n, s n+1 ) + 2s(s + 1) 8

9 we can express the local density H s n,n+1 via the invariant In particular for s = 1 we get σ = ( s n, s n+1 ). H 1 n,n+1 = σ σ 2. Classical contraction corresponds to the case s and from the asymptotics of ψ-function one can easily get the classical limit (7). Thus starting from the construction of the Hilbert space as a carrier of the representation of the basic variables the spin operators s n we were able to quantize the dynamical model with Hamiltonian H, given by (7), retaining the integrability. The construction used heavily the auxiliary object the matrix L n (λ). In other words, it did not follow from some universal receipt, rather it was based on very particular property of the classical system. Of course, the concrete system, considered here is a particular example of a large family of models, which appeared in connection with the Inverse Scattering Method. References [1] L. D. Faddeev, L. A. Takhtajan, Hamiltonian Methods in the Theory of Solitons, Springer-Verlag 1987 (Springer series in Soviet Mathematics). [2] L. D. Faddeev, How algrbraic Bethe ansatz works for integrable models, proceedings of Les Houches School of Physics: Quantum Symmetries, Les Houches 1995, [3] M. Flato, A Short Survey in Modern Statistical Mechanics, in Istanbul 1970, Studies In Mathematical Physics, Dordrecht 1973,

Quantum Mechanics and Representation Theory

Quantum Mechanics and Representation Theory Quantum Mechanics and Representation Theory Peter Woit Columbia University Texas Tech, November 21 2013 Peter Woit (Columbia University) Quantum Mechanics and Representation Theory November 2013 1 / 30

More information

1D 3D 1D 3D. is called eigenstate or state function. When an operator act on a state, it can be written as

1D 3D 1D 3D. is called eigenstate or state function. When an operator act on a state, it can be written as Chapter 3 (Lecture 4-5) Postulates of Quantum Mechanics Now we turn to an application of the preceding material, and move into the foundations of quantum mechanics. Quantum mechanics is based on a series

More information

Chapter 9 Unitary Groups and SU(N)

Chapter 9 Unitary Groups and SU(N) Chapter 9 Unitary Groups and SU(N) The irreducible representations of SO(3) are appropriate for describing the degeneracies of states of quantum mechanical systems which have rotational symmetry in three

More information

SURVEY ON DYNAMICAL YANG-BAXTER MAPS. Y. Shibukawa

SURVEY ON DYNAMICAL YANG-BAXTER MAPS. Y. Shibukawa SURVEY ON DYNAMICAL YANG-BAXTER MAPS Y. Shibukawa Department of Mathematics, Faculty of Science, Hokkaido University, Sapporo 060-0810, Japan e-mail: shibu@math.sci.hokudai.ac.jp Abstract In this survey,

More information

Three Pictures of Quantum Mechanics. Thomas R. Shafer April 17, 2009

Three Pictures of Quantum Mechanics. Thomas R. Shafer April 17, 2009 Three Pictures of Quantum Mechanics Thomas R. Shafer April 17, 2009 Outline of the Talk Brief review of (or introduction to) quantum mechanics. 3 different viewpoints on calculation. Schrödinger, Heisenberg,

More information

DETERMINANTS. b 2. x 2

DETERMINANTS. b 2. x 2 DETERMINANTS 1 Systems of two equations in two unknowns A system of two equations in two unknowns has the form a 11 x 1 + a 12 x 2 = b 1 a 21 x 1 + a 22 x 2 = b 2 This can be written more concisely in

More information

H = = + H (2) And thus these elements are zero. Now we can try to do the same for time reversal. Remember the

H = = + H (2) And thus these elements are zero. Now we can try to do the same for time reversal. Remember the 1 INTRODUCTION 1 1. Introduction In the discussion of random matrix theory and information theory, we basically explained the statistical aspect of ensembles of random matrices, The minimal information

More information

1 Variational calculation of a 1D bound state

1 Variational calculation of a 1D bound state TEORETISK FYSIK, KTH TENTAMEN I KVANTMEKANIK FÖRDJUPNINGSKURS EXAMINATION IN ADVANCED QUANTUM MECHAN- ICS Kvantmekanik fördjupningskurs SI38 för F4 Thursday December, 7, 8. 13. Write on each page: Name,

More information

Hermitian Operators An important property of operators is suggested by considering the Hamiltonian for the particle in a box: d 2 dx 2 (1)

Hermitian Operators An important property of operators is suggested by considering the Hamiltonian for the particle in a box: d 2 dx 2 (1) CHAPTER 4 PRINCIPLES OF QUANTUM MECHANICS In this Chapter we will continue to develop the mathematical formalism of quantum mechanics, using heuristic arguments as necessary. This will lead to a system

More information

Finite dimensional C -algebras

Finite dimensional C -algebras Finite dimensional C -algebras S. Sundar September 14, 2012 Throughout H, K stand for finite dimensional Hilbert spaces. 1 Spectral theorem for self-adjoint opertors Let A B(H) and let {ξ 1, ξ 2,, ξ n

More information

Mathematical Physics, Lecture 9

Mathematical Physics, Lecture 9 Mathematical Physics, Lecture 9 Hoshang Heydari Fysikum April 25, 2012 Hoshang Heydari (Fysikum) Mathematical Physics, Lecture 9 April 25, 2012 1 / 42 Table of contents 1 Differentiable manifolds 2 Differential

More information

We consider a hydrogen atom in the ground state in the uniform electric field

We consider a hydrogen atom in the ground state in the uniform electric field Lecture 13 Page 1 Lectures 13-14 Hydrogen atom in electric field. Quadratic Stark effect. Atomic polarizability. Emission and Absorption of Electromagnetic Radiation by Atoms Transition probabilities and

More information

Similarity and Diagonalization. Similar Matrices

Similarity and Diagonalization. Similar Matrices MATH022 Linear Algebra Brief lecture notes 48 Similarity and Diagonalization Similar Matrices Let A and B be n n matrices. We say that A is similar to B if there is an invertible n n matrix P such that

More information

CLASSIFICATIONS OF STAR PRODUCTS AND DEFORMATIONS OF POISSON BRACKETS

CLASSIFICATIONS OF STAR PRODUCTS AND DEFORMATIONS OF POISSON BRACKETS POISSON GEOMETRY BANACH CENTER PUBLICATIONS, VOLUME 51 INSTITUTE OF MATHEMATICS POLISH ACADEMY OF SCIENCES WARSZAWA 000 CLASSIFICATIONS OF STAR PRODUCTS AND DEFORMATIONS OF POISSON BRACKETS PHILIPP E BO

More information

Lecture 22 Relevant sections in text: 3.1, 3.2. Rotations in quantum mechanics

Lecture 22 Relevant sections in text: 3.1, 3.2. Rotations in quantum mechanics Lecture Relevant sections in text: 3.1, 3. Rotations in quantum mechanics Now we will discuss what the preceding considerations have to do with quantum mechanics. In quantum mechanics transformations in

More information

LECTURE III. Bi-Hamiltonian chains and it projections. Maciej B laszak. Poznań University, Poland

LECTURE III. Bi-Hamiltonian chains and it projections. Maciej B laszak. Poznań University, Poland LECTURE III Bi-Hamiltonian chains and it projections Maciej B laszak Poznań University, Poland Maciej B laszak (Poznań University, Poland) LECTURE III 1 / 18 Bi-Hamiltonian chains Let (M, Π) be a Poisson

More information

Quantum Field Theory and Representation Theory

Quantum Field Theory and Representation Theory Quantum Field Theory and Representation Theory Peter Woit woit@math.columbia.edu Department of Mathematics Columbia University Quantum Field Theory and Representation Theory p.1 Outline of the talk Quantum

More information

THE DIMENSION OF A VECTOR SPACE

THE DIMENSION OF A VECTOR SPACE THE DIMENSION OF A VECTOR SPACE KEITH CONRAD This handout is a supplementary discussion leading up to the definition of dimension and some of its basic properties. Let V be a vector space over a field

More information

Tensors on a vector space

Tensors on a vector space APPENDIX B Tensors on a vector space In this Appendix, we gather mathematical definitions and results pertaining to tensors. The purpose is mostly to introduce the modern, geometrical view on tensors,

More information

MASTER OF SCIENCE IN PHYSICS MASTER OF SCIENCES IN PHYSICS (MS PHYS) (LIST OF COURSES BY SEMESTER, THESIS OPTION)

MASTER OF SCIENCE IN PHYSICS MASTER OF SCIENCES IN PHYSICS (MS PHYS) (LIST OF COURSES BY SEMESTER, THESIS OPTION) MASTER OF SCIENCE IN PHYSICS Admission Requirements 1. Possession of a BS degree from a reputable institution or, for non-physics majors, a GPA of 2.5 or better in at least 15 units in the following advanced

More information

Introduction to A Quantum Theory over A Galois Field

Introduction to A Quantum Theory over A Galois Field Symmetry 2010, 2, 1810-1845; doi:10.3390/sym2041810 OPEN ACCESS symmetry ISSN 2073-8994 www.mdpi.com/journal/symmetry Article Introduction to A Quantum Theory over A Galois Field Felix M. Lev Artwork Conversion

More information

Hilbert Space Quantum Mechanics

Hilbert Space Quantum Mechanics qitd114 Hilbert Space Quantum Mechanics Robert B. Griffiths Version of 16 January 2014 Contents 1 Introduction 1 1.1 Hilbert space............................................. 1 1.2 Qubit.................................................

More information

Elasticity Theory Basics

Elasticity Theory Basics G22.3033-002: Topics in Computer Graphics: Lecture #7 Geometric Modeling New York University Elasticity Theory Basics Lecture #7: 20 October 2003 Lecturer: Denis Zorin Scribe: Adrian Secord, Yotam Gingold

More information

P.A.M. Dirac Received May 29, 1931

P.A.M. Dirac Received May 29, 1931 P.A.M. Dirac, Proc. Roy. Soc. A 133, 60 1931 Quantised Singularities in the Electromagnetic Field P.A.M. Dirac Received May 29, 1931 1. Introduction The steady progress of physics requires for its theoretical

More information

Particle Physics. Michaelmas Term 2011 Prof Mark Thomson. Handout 7 : Symmetries and the Quark Model. Introduction/Aims

Particle Physics. Michaelmas Term 2011 Prof Mark Thomson. Handout 7 : Symmetries and the Quark Model. Introduction/Aims Particle Physics Michaelmas Term 2011 Prof Mark Thomson Handout 7 : Symmetries and the Quark Model Prof. M.A. Thomson Michaelmas 2011 206 Introduction/Aims Symmetries play a central role in particle physics;

More information

Topologically Massive Gravity with a Cosmological Constant

Topologically Massive Gravity with a Cosmological Constant Topologically Massive Gravity with a Cosmological Constant Derek K. Wise Joint work with S. Carlip, S. Deser, A. Waldron Details and references at arxiv:0803.3998 [hep-th] (or for the short story, 0807.0486,

More information

University of Lille I PC first year list of exercises n 7. Review

University of Lille I PC first year list of exercises n 7. Review University of Lille I PC first year list of exercises n 7 Review Exercise Solve the following systems in 4 different ways (by substitution, by the Gauss method, by inverting the matrix of coefficients

More information

3. Derive the partition function for the ideal monoatomic gas. Use Boltzmann statistics, and a quantum mechanical model for the gas.

3. Derive the partition function for the ideal monoatomic gas. Use Boltzmann statistics, and a quantum mechanical model for the gas. Tentamen i Statistisk Fysik I den tjugosjunde februari 2009, under tiden 9.00-15.00. Lärare: Ingemar Bengtsson. Hjälpmedel: Penna, suddgummi och linjal. Bedömning: 3 poäng/uppgift. Betyg: 0-3 = F, 4-6

More information

Lecture 2. Observables

Lecture 2. Observables Lecture 2 Observables 13 14 LECTURE 2. OBSERVABLES 2.1 Observing observables We have seen at the end of the previous lecture that each dynamical variable is associated to a linear operator Ô, and its expectation

More information

Lecture L3 - Vectors, Matrices and Coordinate Transformations

Lecture L3 - Vectors, Matrices and Coordinate Transformations S. Widnall 16.07 Dynamics Fall 2009 Lecture notes based on J. Peraire Version 2.0 Lecture L3 - Vectors, Matrices and Coordinate Transformations By using vectors and defining appropriate operations between

More information

NOTES ON LINEAR TRANSFORMATIONS

NOTES ON LINEAR TRANSFORMATIONS NOTES ON LINEAR TRANSFORMATIONS Definition 1. Let V and W be vector spaces. A function T : V W is a linear transformation from V to W if the following two properties hold. i T v + v = T v + T v for all

More information

MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS. + + x 2. x n. a 11 a 12 a 1n b 1 a 21 a 22 a 2n b 2 a 31 a 32 a 3n b 3. a m1 a m2 a mn b m

MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS. + + x 2. x n. a 11 a 12 a 1n b 1 a 21 a 22 a 2n b 2 a 31 a 32 a 3n b 3. a m1 a m2 a mn b m MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS 1. SYSTEMS OF EQUATIONS AND MATRICES 1.1. Representation of a linear system. The general system of m equations in n unknowns can be written a 11 x 1 + a 12 x 2 +

More information

State of Stress at Point

State of Stress at Point State of Stress at Point Einstein Notation The basic idea of Einstein notation is that a covector and a vector can form a scalar: This is typically written as an explicit sum: According to this convention,

More information

Special Theory of Relativity

Special Theory of Relativity June 1, 2010 1 1 J.D.Jackson, Classical Electrodynamics, 3rd Edition, Chapter 11 Introduction Einstein s theory of special relativity is based on the assumption (which might be a deep-rooted superstition

More information

Classical Physics Prof. V. Balakrishnan Department of Physics Indian Institution of Technology, Madras. Lecture No. # 13

Classical Physics Prof. V. Balakrishnan Department of Physics Indian Institution of Technology, Madras. Lecture No. # 13 Classical Physics Prof. V. Balakrishnan Department of Physics Indian Institution of Technology, Madras Lecture No. # 13 Now, let me formalize the idea of symmetry, what I mean by symmetry, what we mean

More information

Lecture 3 SU(2) January 26, 2011. Lecture 3

Lecture 3 SU(2) January 26, 2011. Lecture 3 Lecture 3 SU(2) January 26, 2 Lecture 3 A Little Group Theory A group is a set of elements plus a compostion rule, such that:. Combining two elements under the rule gives another element of the group.

More information

PHY4604 Introduction to Quantum Mechanics Fall 2004 Practice Test 3 November 22, 2004

PHY4604 Introduction to Quantum Mechanics Fall 2004 Practice Test 3 November 22, 2004 PHY464 Introduction to Quantum Mechanics Fall 4 Practice Test 3 November, 4 These problems are similar but not identical to the actual test. One or two parts will actually show up.. Short answer. (a) Recall

More information

Basic Concepts in Nuclear Physics

Basic Concepts in Nuclear Physics Basic Concepts in Nuclear Physics Paolo Finelli Corso di Teoria delle Forze Nucleari 2011 Literature/Bibliography Some useful texts are available at the Library: Wong, Nuclear Physics Krane, Introductory

More information

INTERACTION OF TWO CHARGES IN A UNIFORM MAGNETIC FIELD: II. SPATIAL PROBLEM

INTERACTION OF TWO CHARGES IN A UNIFORM MAGNETIC FIELD: II. SPATIAL PROBLEM INTERACTION OF TWO CHARGES IN A UNIFORM MAGNETIC FIELD: II. SPATIAL PROBLEM D. PINHEIRO AND R. S. MACKAY Dedicated to the memory of John Greene. Abstract. The interaction of two charges moving in R 3 in

More information

For the case of an N-dimensional spinor the vector α is associated to the onedimensional . N

For the case of an N-dimensional spinor the vector α is associated to the onedimensional . N 1 CHAPTER 1 Review of basic Quantum Mechanics concepts Introduction. Hermitian operators. Physical meaning of the eigenvectors and eigenvalues of Hermitian operators. Representations and their use. on-hermitian

More information

Lecture 2: Essential quantum mechanics

Lecture 2: Essential quantum mechanics Department of Physical Sciences, University of Helsinki http://theory.physics.helsinki.fi/ kvanttilaskenta/ p. 1/46 Quantum information and computing Lecture 2: Essential quantum mechanics Jani-Petri Martikainen

More information

arxiv:hep-th/9702066v1 7 Feb 1997

arxiv:hep-th/9702066v1 7 Feb 1997 Oslo SHS-96-9, OSLO-TP 4-97 hep-th/9702066 Algebra of Observables for Identical Particles in One Dimension arxiv:hep-th/9702066v 7 Feb 997 Serguei B. Isakov a,b, Jon Magne Leinaas a,b, Jan Myrheim a,c,

More information

8.04: Quantum Mechanics Professor Allan Adams Massachusetts Institute of Technology. Problem Set 5

8.04: Quantum Mechanics Professor Allan Adams Massachusetts Institute of Technology. Problem Set 5 8.04: Quantum Mechanics Professor Allan Adams Massachusetts Institute of Technology Tuesday March 5 Problem Set 5 Due Tuesday March 12 at 11.00AM Assigned Reading: E&R 6 9, App-I Li. 7 1 4 Ga. 4 7, 6 1,2

More information

The Quantum Harmonic Oscillator Stephen Webb

The Quantum Harmonic Oscillator Stephen Webb The Quantum Harmonic Oscillator Stephen Webb The Importance of the Harmonic Oscillator The quantum harmonic oscillator holds a unique importance in quantum mechanics, as it is both one of the few problems

More information

Techniques algébriques en calcul quantique

Techniques algébriques en calcul quantique Techniques algébriques en calcul quantique E. Jeandel Laboratoire de l Informatique du Parallélisme LIP, ENS Lyon, CNRS, INRIA, UCB Lyon 8 Avril 25 E. Jeandel, LIP, ENS Lyon Techniques algébriques en calcul

More information

Is Quantum Mechanics Exact?

Is Quantum Mechanics Exact? Is Quantum Mechanics Exact? Anton Kapustin Simons Center for Geometry and Physics Stony Brook University This year Quantum Theory will celebrate its 90th birthday. Werner Heisenberg s paper Quantum theoretic

More information

Definition 12 An alternating bilinear form on a vector space V is a map B : V V F such that

Definition 12 An alternating bilinear form on a vector space V is a map B : V V F such that 4 Exterior algebra 4.1 Lines and 2-vectors The time has come now to develop some new linear algebra in order to handle the space of lines in a projective space P (V ). In the projective plane we have seen

More information

8.1 Examples, definitions, and basic properties

8.1 Examples, definitions, and basic properties 8 De Rham cohomology Last updated: May 21, 211. 8.1 Examples, definitions, and basic properties A k-form ω Ω k (M) is closed if dω =. It is exact if there is a (k 1)-form σ Ω k 1 (M) such that dσ = ω.

More information

1 Scalars, Vectors and Tensors

1 Scalars, Vectors and Tensors DEPARTMENT OF PHYSICS INDIAN INSTITUTE OF TECHNOLOGY, MADRAS PH350 Classical Physics Handout 1 8.8.2009 1 Scalars, Vectors and Tensors In physics, we are interested in obtaining laws (in the form of mathematical

More information

2. Introduction to quantum mechanics

2. Introduction to quantum mechanics 2. Introduction to quantum mechanics 2.1 Linear algebra Dirac notation Complex conjugate Vector/ket Dual vector/bra Inner product/bracket Tensor product Complex conj. matrix Transpose of matrix Hermitian

More information

Mathematics Course 111: Algebra I Part IV: Vector Spaces

Mathematics Course 111: Algebra I Part IV: Vector Spaces Mathematics Course 111: Algebra I Part IV: Vector Spaces D. R. Wilkins Academic Year 1996-7 9 Vector Spaces A vector space over some field K is an algebraic structure consisting of a set V on which are

More information

3. A LITTLE ABOUT GROUP THEORY

3. A LITTLE ABOUT GROUP THEORY 3. A LITTLE ABOUT GROUP THEORY 3.1 Preliminaries It is an apparent fact that nature exhibits many symmetries, both exact and approximate. A symmetry is an invariance property of a system under a set of

More information

Matrix Representations of Linear Transformations and Changes of Coordinates

Matrix Representations of Linear Transformations and Changes of Coordinates Matrix Representations of Linear Transformations and Changes of Coordinates 01 Subspaces and Bases 011 Definitions A subspace V of R n is a subset of R n that contains the zero element and is closed under

More information

RESONANCES AND BALLS IN OBSTACLE SCATTERING WITH NEUMANN BOUNDARY CONDITIONS

RESONANCES AND BALLS IN OBSTACLE SCATTERING WITH NEUMANN BOUNDARY CONDITIONS RESONANCES AND BALLS IN OBSTACLE SCATTERING WITH NEUMANN BOUNDARY CONDITIONS T. J. CHRISTIANSEN Abstract. We consider scattering by an obstacle in R d, d 3 odd. We show that for the Neumann Laplacian if

More information

A Concise Text on Advanced Linear Algebra

A Concise Text on Advanced Linear Algebra A Concise Text on Advanced Linear Algebra This engaging textbook for advanced undergraduate students and beginning graduates covers the core subjects in linear algebra. The author motivates the concepts

More information

LQG with all the degrees of freedom

LQG with all the degrees of freedom LQG with all the degrees of freedom Marcin Domagała, Kristina Giesel, Wojciech Kamiński, Jerzy Lewandowski arxiv:1009.2445 LQG with all the degrees of freedom p.1 Quantum gravity within reach The recent

More information

MICROLOCAL ANALYSIS OF THE BOCHNER-MARTINELLI INTEGRAL

MICROLOCAL ANALYSIS OF THE BOCHNER-MARTINELLI INTEGRAL PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 00, Number 0, Pages 000 000 S 0002-9939(XX)0000-0 MICROLOCAL ANALYSIS OF THE BOCHNER-MARTINELLI INTEGRAL NIKOLAI TARKHANOV AND NIKOLAI VASILEVSKI

More information

1 Monday, October 24: Special Relativity Review

1 Monday, October 24: Special Relativity Review 1 Monday, October 24: Special Relativity Review Personally, I m very fond of classical physics. As we ve seen recently, you can derive some very useful electromagnetic formulae without taking into account

More information

MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS

MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS Systems of Equations and Matrices Representation of a linear system The general system of m equations in n unknowns can be written a x + a 2 x 2 + + a n x n b a

More information

The Dirichlet Unit Theorem

The Dirichlet Unit Theorem Chapter 6 The Dirichlet Unit Theorem As usual, we will be working in the ring B of algebraic integers of a number field L. Two factorizations of an element of B are regarded as essentially the same if

More information

Operator methods in quantum mechanics

Operator methods in quantum mechanics Chapter 3 Operator methods in quantum mechanics While the wave mechanical formulation has proved successful in describing the quantum mechanics of bound and unbound particles, some properties can not be

More information

1 Lecture 3: Operators in Quantum Mechanics

1 Lecture 3: Operators in Quantum Mechanics 1 Lecture 3: Operators in Quantum Mechanics 1.1 Basic notions of operator algebra. In the previous lectures we have met operators: ˆx and ˆp = i h they are called fundamental operators. Many operators

More information

Quadratic Equations in Finite Fields of Characteristic 2

Quadratic Equations in Finite Fields of Characteristic 2 Quadratic Equations in Finite Fields of Characteristic 2 Klaus Pommerening May 2000 english version February 2012 Quadratic equations over fields of characteristic 2 are solved by the well known quadratic

More information

1 Introduction to Matrices

1 Introduction to Matrices 1 Introduction to Matrices In this section, important definitions and results from matrix algebra that are useful in regression analysis are introduced. While all statements below regarding the columns

More information

arxiv:0706.1300v1 [q-fin.pr] 9 Jun 2007

arxiv:0706.1300v1 [q-fin.pr] 9 Jun 2007 THE QUANTUM BLACK-SCHOLES EQUATION LUIGI ACCARDI AND ANDREAS BOUKAS arxiv:0706.1300v1 [q-fin.pr] 9 Jun 2007 Abstract. Motivated by the work of Segal and Segal in [16] on the Black-Scholes pricing formula

More information

Quanta of Geometry and Unification

Quanta of Geometry and Unification Quanta of Geometry and Unification Memorial Meeting for Abdus Salam 90 th Birthday Ali Chamseddine American University of Beirut (AUB) and Institut des Hautes Etudes Scientifique (IHES) January 25-28,

More information

Introduction to Matrix Algebra

Introduction to Matrix Algebra Psychology 7291: Multivariate Statistics (Carey) 8/27/98 Matrix Algebra - 1 Introduction to Matrix Algebra Definitions: A matrix is a collection of numbers ordered by rows and columns. It is customary

More information

CHAPTER 12 MOLECULAR SYMMETRY

CHAPTER 12 MOLECULAR SYMMETRY CHAPTER 12 MOLECULAR SYMMETRY In many cases, the symmetry of a molecule provides a great deal of information about its quantum states, even without a detailed solution of the Schrödinger equation. A geometrical

More information

Math 225A, Differential Topology: Homework 3

Math 225A, Differential Topology: Homework 3 Math 225A, Differential Topology: Homework 3 Ian Coley October 17, 2013 Problem 1.4.7. Suppose that y is a regular value of f : X Y, where X is compact and dim X = dim Y. Show that f 1 (y) is a finite

More information

T ( a i x i ) = a i T (x i ).

T ( a i x i ) = a i T (x i ). Chapter 2 Defn 1. (p. 65) Let V and W be vector spaces (over F ). We call a function T : V W a linear transformation form V to W if, for all x, y V and c F, we have (a) T (x + y) = T (x) + T (y) and (b)

More information

Quantum Mechanics I. Peter S. Riseborough. August 29, 2013

Quantum Mechanics I. Peter S. Riseborough. August 29, 2013 Quantum Mechanics I Peter S. Riseborough August 9, 3 Contents Principles of Classical Mechanics 9. Lagrangian Mechanics........................ 9.. Exercise............................. Solution.............................3

More information

Second postulate of Quantum mechanics: If a system is in a quantum state represented by a wavefunction ψ, then 2

Second postulate of Quantum mechanics: If a system is in a quantum state represented by a wavefunction ψ, then 2 . POSTULATES OF QUANTUM MECHANICS. Introducing the state function Quantum physicists are interested in all kinds of physical systems (photons, conduction electrons in metals and semiconductors, atoms,

More information

Elements of Abstract Group Theory

Elements of Abstract Group Theory Chapter 2 Elements of Abstract Group Theory Mathematics is a game played according to certain simple rules with meaningless marks on paper. David Hilbert The importance of symmetry in physics, and for

More information

PROJECTIVE GEOMETRY. b3 course 2003. Nigel Hitchin

PROJECTIVE GEOMETRY. b3 course 2003. Nigel Hitchin PROJECTIVE GEOMETRY b3 course 2003 Nigel Hitchin hitchin@maths.ox.ac.uk 1 1 Introduction This is a course on projective geometry. Probably your idea of geometry in the past has been based on triangles

More information

Harmonic Oscillator and Coherent States

Harmonic Oscillator and Coherent States Chapter 5 Harmonic Oscillator and Coherent States 5. Harmonic Oscillator In this chapter we will study the features of one of the most important potentials in physics, it s the harmonic oscillator potential

More information

Basic Quantum Mechanics

Basic Quantum Mechanics Basic Quantum Mechanics Postulates of QM - The state of a system with n position variables q, q, qn is specified by a state (or wave) function Ψ(q, q, qn) - To every observable (physical magnitude) there

More information

- develop a theory that describes the wave properties of particles correctly

- develop a theory that describes the wave properties of particles correctly Quantum Mechanics Bohr's model: BUT: In 1925-26: by 1930s: - one of the first ones to use idea of matter waves to solve a problem - gives good explanation of spectrum of single electron atoms, like hydrogen

More information

The three-dimensional rotations are defined as the linear transformations of the vector x = (x 1, x 2, x 3 ) x i = R ij x j, (2.1) x 2 = x 2. (2.

The three-dimensional rotations are defined as the linear transformations of the vector x = (x 1, x 2, x 3 ) x i = R ij x j, (2.1) x 2 = x 2. (2. 2 The rotation group In this Chapter we give a short account of the main properties of the threedimensional rotation group SO(3) and of its universal covering group SU(2). The group SO(3) is an important

More information

Contents. Gbur, Gregory J. Mathematical methods for optical physics and engineering digitalisiert durch: IDS Basel Bern

Contents. Gbur, Gregory J. Mathematical methods for optical physics and engineering digitalisiert durch: IDS Basel Bern Preface page xv 1 Vector algebra 1 1.1 Preliminaries 1 1.2 Coordinate System invariance 4 1.3 Vector multiplication 9 1.4 Useful products of vectors 12 1.5 Linear vector Spaces 13 1.6 Focus: periodic media

More information

Continuous Groups, Lie Groups, and Lie Algebras

Continuous Groups, Lie Groups, and Lie Algebras Chapter 7 Continuous Groups, Lie Groups, and Lie Algebras Zeno was concerned with three problems... These are the problem of the infinitesimal, the infinite, and continuity... Bertrand Russell The groups

More information

Associativity condition for some alternative algebras of degree three

Associativity condition for some alternative algebras of degree three Associativity condition for some alternative algebras of degree three Mirela Stefanescu and Cristina Flaut Abstract In this paper we find an associativity condition for a class of alternative algebras

More information

The Schrödinger Equation

The Schrödinger Equation The Schrödinger Equation When we talked about the axioms of quantum mechanics, we gave a reduced list. We did not talk about how to determine the eigenfunctions for a given situation, or the time development

More information

Numerical Analysis Lecture Notes

Numerical Analysis Lecture Notes Numerical Analysis Lecture Notes Peter J. Olver 6. Eigenvalues and Singular Values In this section, we collect together the basic facts about eigenvalues and eigenvectors. From a geometrical viewpoint,

More information

202 6TheMechanicsofQuantumMechanics. ) ψ (r) =± φ (r) ψ (r) (6.139) (t)

202 6TheMechanicsofQuantumMechanics. ) ψ (r) =± φ (r) ψ (r) (6.139) (t) 0 6TheMechanicsofQuantumMechanics It is easily shown that ˆ has only two (real) eigenvalues, ±1. That is, if ˆ ψ (r) = λψ (r), ˆ ψ (r) = ˆ ψ ( r) = ψ (r) = λ ˆ ψ ( r) = λ ψ (r) (6.137) so λ =±1. Thus,

More information

Arithmetic complexity in algebraic extensions

Arithmetic complexity in algebraic extensions Arithmetic complexity in algebraic extensions Pavel Hrubeš Amir Yehudayoff Abstract Given a polynomial f with coefficients from a field F, is it easier to compute f over an extension R of F than over F?

More information

The Characteristic Polynomial

The Characteristic Polynomial Physics 116A Winter 2011 The Characteristic Polynomial 1 Coefficients of the characteristic polynomial Consider the eigenvalue problem for an n n matrix A, A v = λ v, v 0 (1) The solution to this problem

More information

Lecture No. # 02 Prologue-Part 2

Lecture No. # 02 Prologue-Part 2 Advanced Matrix Theory and Linear Algebra for Engineers Prof. R.Vittal Rao Center for Electronics Design and Technology Indian Institute of Science, Bangalore Lecture No. # 02 Prologue-Part 2 In the last

More information

Entangling rates and the quantum holographic butterfly

Entangling rates and the quantum holographic butterfly Entangling rates and the quantum holographic butterfly David Berenstein DAMTP/UCSB. C. Asplund, D.B. arxiv:1503.04857 Work in progress with C. Asplund, A. Garcia Garcia Questions If a black hole is in

More information

Integrable probability: Macdonald processes, quantum integrable systems and the Kardar-Parisi-Zhang universality class

Integrable probability: Macdonald processes, quantum integrable systems and the Kardar-Parisi-Zhang universality class Tueday talk 2b Page 1 Integrable probability: Macdonald processes, quantum integrable systems and the Kardar-Parisi-Zhang universality class Ivan Corwin (Columbia University, Clay Mathematics Institute,

More information

General theory of stochastic processes

General theory of stochastic processes CHAPTER 1 General theory of stochastic processes 1.1. Definition of stochastic process First let us recall the definition of a random variable. A random variable is a random number appearing as a result

More information

Parabolic Equations. Chapter 5. Contents. 5.1.2 Well-Posed Initial-Boundary Value Problem. 5.1.3 Time Irreversibility of the Heat Equation

Parabolic Equations. Chapter 5. Contents. 5.1.2 Well-Posed Initial-Boundary Value Problem. 5.1.3 Time Irreversibility of the Heat Equation 7 5.1 Definitions Properties Chapter 5 Parabolic Equations Note that we require the solution u(, t bounded in R n for all t. In particular we assume that the boundedness of the smooth function u at infinity

More information

ORDERS OF ELEMENTS IN A GROUP

ORDERS OF ELEMENTS IN A GROUP ORDERS OF ELEMENTS IN A GROUP KEITH CONRAD 1. Introduction Let G be a group and g G. We say g has finite order if g n = e for some positive integer n. For example, 1 and i have finite order in C, since

More information

Statistical Physics, Part 2 by E. M. Lifshitz and L. P. Pitaevskii (volume 9 of Landau and Lifshitz, Course of Theoretical Physics).

Statistical Physics, Part 2 by E. M. Lifshitz and L. P. Pitaevskii (volume 9 of Landau and Lifshitz, Course of Theoretical Physics). Fermi liquids The electric properties of most metals can be well understood from treating the electrons as non-interacting. This free electron model describes the electrons in the outermost shell of the

More information

Magic Squares and Syzygies Richard P. Stanley

Magic Squares and Syzygies Richard P. Stanley Magic Squares and Syzygies p. 1 Magic Squares and Syzygies Richard P. Stanley Magic Squares and Syzygies p. 2 Traditional magic squares Many elegant, ingenious, and beautiful constructions, but no general

More information

The derivation of the balance equations

The derivation of the balance equations Chapter 3 The derivation of the balance equations In this chapter we present the derivation of the balance equations for an arbitrary physical quantity which starts from the Liouville equation. We follow,

More information

Matrices, Determinants and Linear Systems

Matrices, Determinants and Linear Systems September 21, 2014 Matrices A matrix A m n is an array of numbers in rows and columns a 11 a 12 a 1n r 1 a 21 a 22 a 2n r 2....... a m1 a m2 a mn r m c 1 c 2 c n We say that the dimension of A is m n (we

More information

Laws of Motion and Conservation Laws

Laws of Motion and Conservation Laws Laws of Motion and Conservation Laws The first astrophysics we ll consider will be gravity, which we ll address in the next class. First, though, we need to set the stage by talking about some of the basic

More information

4: EIGENVALUES, EIGENVECTORS, DIAGONALIZATION

4: EIGENVALUES, EIGENVECTORS, DIAGONALIZATION 4: EIGENVALUES, EIGENVECTORS, DIAGONALIZATION STEVEN HEILMAN Contents 1. Review 1 2. Diagonal Matrices 1 3. Eigenvectors and Eigenvalues 2 4. Characteristic Polynomial 4 5. Diagonalizability 6 6. Appendix:

More information

Linear Algebra Review. Vectors

Linear Algebra Review. Vectors Linear Algebra Review By Tim K. Marks UCSD Borrows heavily from: Jana Kosecka kosecka@cs.gmu.edu http://cs.gmu.edu/~kosecka/cs682.html Virginia de Sa Cogsci 8F Linear Algebra review UCSD Vectors The length

More information

4. Matrix inverses. left and right inverse. linear independence. nonsingular matrices. matrices with linearly independent columns

4. Matrix inverses. left and right inverse. linear independence. nonsingular matrices. matrices with linearly independent columns L. Vandenberghe EE133A (Spring 2016) 4. Matrix inverses left and right inverse linear independence nonsingular matrices matrices with linearly independent columns matrices with linearly independent rows

More information