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Preprint typeset in JHEP style - HYPER VERSION Chapter 2: Linear Algebra User's Manual Gregory W. Moore Abstract: An overview of some of the finer points of linear algebra usually omitted in physics courses. April 27, 2018 -TOC- Contents 1. Introduction 5 2. Basic Definitions Of Algebraic Structures: Rings, Fields, Modules, Vec- tor Spaces, And Algebras 6 2.1 Rings 6 2.2 Fields 6 2.3 Modules 8 2.4 Vector Spaces 9 2.5 Algebras 9 3. Linear Transformations 13 4. Basis And Dimension 15 4.1 Linear Independence 15 4.2 Free Modules 16 4.3 Vector Spaces 16 4.4 Linear Operators And Matrices 19 4.5 Determinant And Trace 22 5. New Vector Spaces from Old Ones 23 5.1 Direct sum 23 5.2 Quotient Space 27 5.3 Tensor Product 28 5.4 Dual Space 32 6. Tensor spaces 35 6.1 Totally Symmetric And Antisymmetric Tensors 36 6.2 Algebraic structures associated with tensors 38 6.2.1 An Approach To Noncommutative Geometry 41 7. Kernel, Image, and Cokernel 42 7.1 The index of a linear operator 43 8. A Taste of Homological Algebra 44 8.1 The Euler-Poincar´eprinciple 45 8.2 Chain maps and chain homotopies 46 8.3 Exact sequences of complexes 46 8.4 Left- and right-exactness 47 { 1 { 9. Relations Between Real, Complex, And Quaternionic Vector Spaces 47 9.1 Complex structure on a real vector space 47 9.2 Real Structure On A Complex Vector Space 50 9.2.1 Complex Conjugate Of A Complex Vector Space 52 9.2.2 Complexification 53 9.3 The Quaternions 55 9.4 Quaternionic Structure On A Real Vector Space 64 9.5 Quaternionic Structure On Complex Vector Space 65 9.5.1 Complex Structure On Quaternionic Vector Space 65 9.5.2 Summary 66 9.6 Spaces Of Real, Complex, Quaternionic Structures 66 10. Some Canonical Forms For a Matrix Under Conjugation 70 10.1 What is a canonical form? 70 10.2 Rank 70 10.3 Eigenvalues and Eigenvectors 71 10.4 Jordan Canonical Form 74 10.4.1 Proof of the Jordan canonical form theorem 79 10.5 The stabilizer of a Jordan canonical form 80 10.5.1 Simultaneous diagonalization 82 11. Sesquilinear forms and (anti)-Hermitian forms 84 12. Inner product spaces, normed linear spaces, and bounded operators 86 12.1 Inner product spaces 86 12.2 Normed linear spaces 87 12.3 Bounded linear operators 88 12.4 Constructions with inner product spaces 89 13. Hilbert space 90 14. Banach space 93 15. Projection operators and orthogonal decomposition 94 16. Unitary, Hermitian, and normal operators 97 17. The spectral theorem: Finite Dimensions 100 17.1 Normal and Unitary matrices 102 17.2 Singular value decomposition and Schmidt decomposition 102 17.2.1 Bidiagonalization 102 17.2.2 Application: The Cabbibo-Kobayashi-Maskawa matrix, or, how bidi- agonalization can win you the Nobel Prize 103 17.2.3 Singular value decomposition 105 17.2.4 Schmidt decomposition 106 { 2 { 18. Operators on Hilbert space 107 18.1 Lies my teacher told me 107 18.1.1 Lie 1: The trace is cyclic: 107 18.1.2 Lie 2: Hermitian operators have real eigenvalues 107 18.1.3 Lie 3: Hermitian operators exponentiate to form one-parameter groups of unitary operators 108 18.2 Hellinger-Toeplitz theorem 108 18.3 Spectrum and resolvent 110 18.4 Spectral theorem for bounded self-adjoint operators 115 18.5 Defining the adjoint of an unbounded operator 120 18.6 Spectral Theorem for unbounded self-adjoint operators 122 18.7 Commuting self-adjoint operators 123 18.8 Stone's theorem 124 18.9 Traceclass operators 125 19. The Dirac-von Neumann axioms of quantum mechanics 128 20. Canonical Forms of Antisymmetric, Symmetric, and Orthogonal matri- ces 135 20.1 Pairings and bilinear forms 135 20.1.1 Perfect pairings 135 20.1.2 Vector spaces 136 20.1.3 Choosing a basis 137 20.2 Canonical forms for symmetric matrices 137 20.3 Orthogonal matrices: The real spectral theorem 140 20.4 Canonical forms for antisymmetric matrices 141 20.5 Automorphism Groups of Bilinear and Sesquilinear Forms 142 21. Other canonical forms: Upper triangular, polar, reduced echelon 144 21.1 General upper triangular decomposition 144 21.2 Gram-Schmidt procedure 144 21.2.1 Orthogonal polynomials 145 21.3 Polar decomposition 147 21.4 Reduced Echelon form 149 22. Families of Matrices 149 22.1 Families of projection operators: The theory of vector bundles 149 22.2 Codimension of the space of coinciding eigenvalues 153 22.2.1 Families of complex matrices: Codimension of coinciding character- istic values 154 22.2.2 Orbits 155 22.2.3 Local model near Ssing 156 22.2.4 Families of Hermitian operators 157 22.3 Canonical form of a family in a first order neighborhood 159 { 3 { 22.4 Families of operators and spectral covers 160 22.5 Families of matrices and differential equations 165 22.5.1 The WKB expansion 167 22.5.2 Monodromy representation and Hilbert's 21st problem 170 22.5.3 Stokes' phenomenon 170 23. Z2-graded, or super-, linear algebra 175 23.1 Super vector spaces 175 23.2 Linear transformations between supervector spaces 179 23.3 Superalgebras 181 23.4 Modules over superalgebras 187 23.5 Free modules and the super-General Linear Group 191 23.6 The Supertrace 193 23.7 The Berezinian of a linear transformation 194 23.8 Bilinear forms 198 23.9 Star-structures and super-Hilbert spaces 199 23.9.1 SuperUnitary Group 202 23.10Functions on superspace and supermanifolds 203 23.10.1Philosophical background 203 23.10.2The model superspace Rpjq 205 23.10.3Superdomains 206 23.10.4A few words about sheaves 207 23.10.5Definition of supermanifolds 209 23.10.6Supervector fields and super-differential forms 212 23.11Integration over a superdomain 215 23.12Gaussian Integrals 219 23.12.1Reminder on bosonic Gaussian integrals 219 23.12.2Gaussian integral on a fermionic point: Pfaffians 219 23.12.3Gaussian integral on Rpjq 224 23.12.4Supersymmetric Cancelations 225 23.13References 226 24. Determinant Lines, Pfaffian Lines, Berezinian Lines, and anomalies 227 24.1 The determinant and determinant line of a linear operator in finite dimensions227 24.2 Determinant line of a vector space and of a complex 229 24.3 Abstract defining properties of determinants 231 24.4 Pfaffian Line 231 24.5 Determinants and determinant lines in infinite dimensions 233 24.5.1 Determinants 233 24.5.2 Fredholom Operators 234 24.5.3 The determinant line for a family of Fredholm operators 235 24.5.4 The Quillen norm 236 24.5.5 References 237 { 4 { 24.6 Berezinian of a free module 237 24.7 Brief Comments on fermionic path integrals and anomalies 238 24.7.1 General Considerations 238 24.7.2 Determinant of the one-dimensional Dirac operator 239 24.7.3 A supersymmetric quantum mechanics 241 24.7.4 Real Fermions in one dimension coupled to an orthogonal gauge field 242 24.7.5 The global anomaly when M is not spin 243 24.7.6 References 244 25. Quadratic forms and lattices 244 25.1 Definition 245 25.2 Embedded Lattices 247 25.3 Some Invariants of Lattices 252 25.3.1 The characteristic vector 258 25.3.2 The Gauss-Milgram relation 258 25.4 Self-dual lattices 260 25.4.1 Some classification results 263 25.5 Embeddings of lattices: The Nikulin theorem 267 25.6 References 267 26. Positive definite Quadratic forms 267 27. Quivers and their representations 267 1. Introduction Linear algebra is of course very important in many areas of physics. Among them: 1. Tensor analysis - used in classical mechanics and general relativity. 2. The very formulation of quantum mechanics is based on linear algebra: The states in a physical system are described by \rays" in a projective Hilbert space, and physical observables are identified with Hermitian linear operators on Hilbert space. 3. The realization of symmetry in quantum mechanics is through representation theory of groups which relies heavily on linear algebra. For this reason linear algebra is often taught in physics courses. The problem is that it is often mis-taught. Therefore we are going to make a quick review of basic notions stressing some points not usually emphasized in physics courses. We also want to review the basic canonical forms into which various types matrices can be put. These are very useful when discussing various aspects of matrix groups. { 5 { For more information useful references are Herstein, Jacobsen, Lang, Eisenbud, Com- mutative Algebra Springer GTM 150, Atiyah and MacDonald, Introduction to Commutative Algebra. For an excellent terse summary of homological algebra consult S.I. Gelfand and Yu. I. Manin, Homological Algebra. We will only touch briefly on some aspects of functional analysis - which is crucial to quantum mechanics. The standard reference for physicists is: Reed and Simon, Methods of Modern Mathematical Physics, especially, vol. I. 2. Basic Definitions Of Algebraic Structures: Rings, Fields, Modules, Vec- tor Spaces, And Algebras 2.1 Rings In the previous chapter we talked about groups. We now overlay some extra structure on an abelian group R, with operation + and identity 0, to define what is called a ring. The new structure is a second binary operation (a; b) ! a · b 2 R on elements a; b 2 R. We demand that this operation be associative, a · (b · c) = (a · b) · c, and that it is compatible with the pre-existing additive group law. To be precise, the two operations + and · are compatible in the sense that there is a distributive law: a · (b + c) = a · b + a · c (2.1) (a + b) · c = a · c + b · c (2.2) Remarks 1.
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