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. May 3, 2021 -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 7 2.2.1 Finite Fields 8 2.3 Modules 8 2.4 Vector Spaces 9 2.5 Algebras 10 3. Linear Transformations 14 4. Basis And Dimension 16 4.1 Linear Independence 16 4.2 Free Modules 16 4.3 Vector Spaces 17 4.4 Linear Operators And Matrices 20 4.5 Determinant And Trace 23 5. New Vector Spaces from Old Ones 24 5.1 Direct sum 24 5.2 Quotient Space 28 5.3 Tensor Product 30 5.4 Dual Space 34 6. Tensor spaces 38 6.1 Totally Symmetric And Antisymmetric Tensors 39 6.2 Algebraic structures associated with tensors 44 6.2.1 An Approach To Noncommutative Geometry 47 7. Kernel, Image, and Cokernel 47 7.1 The index of a linear operator 50 8. A Taste of Homological Algebra 51 8.1 The Euler-Poincar´eprinciple 54 8.2 Chain maps and chain homotopies 55 8.3 Exact sequences of complexes 56 8.4 Left- and right-exactness 56 { 1 { 9. Relations Between Real, Complex, And Quaternionic Vector Spaces 59 9.1 Complex structure on a real vector space 59 9.2 Real Structure On A Complex Vector Space 64 9.2.1 Complex Conjugate Of A Complex Vector Space 66 9.2.2 Complexification 67 9.3 The Quaternions 69 9.4 Quaternionic Structure On A Real Vector Space 79 9.5 Quaternionic Structure On Complex Vector Space 79 9.5.1 Complex Structure On Quaternionic Vector Space 81 9.5.2 Summary 81 9.6 Spaces Of Real, Complex, Quaternionic Structures 81 10. Some Canonical Forms For a Matrix Under Conjugation 85 10.1 What is a canonical form? 85 10.2 Rank 86 10.3 Eigenvalues and Eigenvectors 87 10.4 Jordan Canonical Form 89 10.4.1 Proof of the Jordan canonical form theorem 94 10.5 The stabilizer of a Jordan canonical form 96 10.5.1 Simultaneous diagonalization 98 11. Sesquilinear forms and (anti)-Hermitian forms 100 12. Inner product spaces, normed linear spaces, and bounded operators 101 12.1 Inner product spaces 101 12.2 Normed linear spaces 103 12.3 Bounded linear operators 104 12.4 Constructions with inner product spaces 105 13. Hilbert space 106 14. Banach space 109 15. Projection operators and orthogonal decomposition 110 16. Unitary, Hermitian, and normal operators 113 17. The spectral theorem: Finite Dimensions 116 17.1 Normal and Unitary matrices 118 17.2 Singular value decomposition and Schmidt decomposition 118 17.2.1 Bidiagonalization 118 17.2.2 Application: The Cabbibo-Kobayashi-Maskawa matrix, or, how bidi- agonalization can win you the Nobel Prize 119 17.2.3 Singular value decomposition 121 17.2.4 Schmidt decomposition 122 { 2 { 18. Operators on Hilbert space 123 18.1 Lies my teacher told me 123 18.1.1 Lie 1: The trace is cyclic: 123 18.1.2 Lie 2: Hermitian operators have real eigenvalues 123 18.1.3 Lie 3: Hermitian operators exponentiate to form one-parameter groups of unitary operators 124 18.2 Hellinger-Toeplitz theorem 124 18.3 Spectrum and resolvent 126 18.4 Spectral theorem for bounded self-adjoint operators 131 18.5 Defining the adjoint of an unbounded operator 136 18.6 Spectral Theorem for unbounded self-adjoint operators 138 18.7 Commuting self-adjoint operators 139 18.8 Stone's theorem 140 18.9 Traceclass operators 141 19. The Dirac-von Neumann axioms of quantum mechanics 144 20. Canonical Forms of Antisymmetric, Symmetric, and Orthogonal matri- ces 151 20.1 Pairings and bilinear forms 151 20.1.1 Perfect pairings 151 20.1.2 Vector spaces 152 20.1.3 Choosing a basis 153 20.2 Canonical forms for symmetric matrices 153 20.3 Orthogonal matrices: The real spectral theorem 156 20.4 Canonical forms for antisymmetric matrices 157 20.5 Automorphism Groups of Bilinear and Sesquilinear Forms 158 21. Other canonical forms: Upper triangular, polar, reduced echelon 160 21.1 General upper triangular decomposition 160 21.2 Gram-Schmidt procedure 160 21.2.1 Orthogonal polynomials 161 21.3 Polar decomposition 163 21.4 Reduced Echelon form 165 22. Families of Matrices 165 22.1 Families of projection operators: The theory of vector bundles 165 22.2 Codimension of the space of coinciding eigenvalues 169 22.2.1 Families of complex matrices: Codimension of coinciding character- istic values 170 22.2.2 Orbits 171 22.2.3 Local model near Ssing 172 22.2.4 Families of Hermitian operators 173 22.3 Canonical form of a family in a first order neighborhood 175 { 3 { 22.4 Families of operators and spectral covers 176 22.5 Families of matrices and differential equations 181 22.5.1 The WKB expansion 183 22.5.2 Monodromy representation and Hilbert's 21st problem 186 22.5.3 Stokes' phenomenon 186 23. Z2-graded, or super-, linear algebra 191 23.1 Super vector spaces 191 23.2 Linear transformations between supervector spaces 195 23.3 Superalgebras 197 23.4 Modules over superalgebras 203 23.5 Free modules and the super-General Linear Group 207 23.6 The Supertrace 209 23.7 The Berezinian of a linear transformation 210 23.8 Bilinear forms 214 23.9 Star-structures and super-Hilbert spaces 215 23.9.1 SuperUnitary Group 218 23.10Functions on superspace and supermanifolds 219 23.10.1Philosophical background 219 23.10.2The model superspace Rpjq 221 23.10.3Superdomains 222 23.10.4A few words about sheaves 223 23.10.5Definition of supermanifolds 225 23.10.6Supervector fields and super-differential forms 228 23.11Integration over a superdomain 231 23.12Gaussian Integrals 235 23.12.1Reminder on bosonic Gaussian integrals 235 23.12.2Gaussian integral on a fermionic point: Pfaffians 235 23.12.3Gaussian integral on Rpjq 240 23.12.4Supersymmetric Cancelations 241 23.13References 242 24. Determinant Lines, Pfaffian Lines, Berezinian Lines, and anomalies 243 24.1 The determinant and determinant line of a linear operator in finite dimensions243 24.2 Determinant line of a vector space and of a complex 245 24.3 Abstract defining properties of determinants 247 24.4 Pfaffian Line 247 24.5 Determinants and determinant lines in infinite dimensions 249 24.5.1 Determinants 249 24.5.2 Fredholom Operators 250 24.5.3 The determinant line for a family of Fredholm operators 251 24.5.4 The Quillen norm 252 24.5.5 References 253 { 4 { 24.6 Berezinian of a free module 253 24.7 Brief Comments on fermionic path integrals and anomalies 254 24.7.1 General Considerations 254 24.7.2 Determinant of the one-dimensional Dirac operator 255 24.7.3 A supersymmetric quantum mechanics 257 24.7.4 Real Fermions in one dimension coupled to an orthogonal gauge field 258 24.7.5 The global anomaly when M is not spin 259 24.7.6 References 260 25. Quadratic Forms And Lattices 260 25.1 Definition 261 25.2 Embedded Lattices 263 25.3 Some Invariants of Lattices 268 25.3.1 The characteristic vector 274 25.3.2 The Gauss-Milgram relation 274 25.4 Self-dual lattices 276 25.4.1 Some classification results 279 25.5 Embeddings of lattices: The Nikulin theorem 283 25.6 References 283 26. Positive definite Quadratic forms 283 27. Quivers and their representations 283 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.