Trace, Metric, and Reality: Notes on Abstract Linear Algebra Adam Coffman Department of Mathematical Sciences Purdue University Fort Wayne 2101 E. Coliseum Blvd. Fort Wayne, IN 46805-1499 Email address: [email protected] URL: http://pfw.edu/math/ 2020 Mathematics Subject Classification. Primary 15-02 Secondary 15A15, 15A63, 15A69, 18M10, 53B30, 53C15 Key words and phrases. Canonical map, linear algebra, trace, Hilbert-Schmidt, indefinite metric, tensor, vector valued form, complex structure operator. This project was supported in part by: A sabbatical semester at Purdue University Fort Wayne in 2020, A sabbatical semester at Indiana University - Purdue University Fort Wayne in 2015, A sabbatical semester at Indiana University - Purdue University Fort Wayne in 2005, A Summer Faculty Research Grant from Indiana University - Purdue University Fort Wayne in 2001, A Summer Faculty Research Grant from the Purdue Research Foundation in 1998. Abstract. Elementary properties of the trace operator, and of some natu- ral vector valued generalizations, are given basis-free statements and proofs, using canonical maps from abstract linear algebra. Properties of contraction with respect to a non-degenerate (but possibly indefinite) metric are similarly analyzed. Several identities are stated geometrically, in terms of the Hilbert- Schmidt trace metric on spaces of linear maps, and metrics induced by tensor products and direct sums. Contents Preface v Overview 1 Chapter 0. Review of Elementary Linear Algebra 3 0.1. Vector spaces 3 0.2. Subspaces 5 0.3. Additive functions and linear functions 7 Chapter 1. Abstract Linear Algebra 9 1.1. Spaces of linear maps 9 1.2. Tensor products 11 1.3. Direct sums 19 1.4. Idempotents and involutions 26 Chapter 2. A Survey of Trace Elements 39 2.1. Endomorphisms: the scalar valued trace 39 2.2. The generalized trace 45 2.3. Vector valued trace 62 2.4. Equivalence of alternative definitions 67 Chapter 3. Bilinear Forms 95 3.1. Metrics 95 3.2. Isometries 99 3.3. Trace with respect to a metric 102 3.4. The Hilbert-Schmidt metric 104 3.5. Orthogonal direct sums 106 3.6. Topics and applications 108 Chapter 4. Vector Valued Bilinear Forms 137 4.1. Transpose for vector valued forms 137 4.2. Symmetric forms 149 4.3. Vector valued trace with respect to a metric 153 4.4. Revisiting the generalized trace 159 4.5. Topics and applications 164 Chapter 5. Complex Structures 175 5.1. Complex Structure Operators 175 5.2. Complex linear and antilinear maps 178 5.3. Commuting Complex Structure Operators 182 5.4. Real trace with complex vector values 212 iii iv CONTENTS Chapter 6. Appendices 215 6.1. Appendix: Quotient spaces 215 6.2. Appendix: Construction of the tensor product 216 Bibliography 219 Preface These notes are a mostly self-contained collection of some theorems of linear algebra that arise in geometry, particularly results about the trace and bilinear forms. Many results are stated with complete proofs, the main method of proof being the use of canonical maps from abstract linear algebra. So, the content of these notes is highly dependent on the notation for these maps developed in Chapter 1. This notation will be used in all the subsequent Chapters, which appear in a logical order, but for 1 <m<n, it is possible to follow Chapter 1 immediately by Chapter n, with only a few citations of Chapter m. To fix notation and review the elementary prerequisites, some foundational material appears in Chapter 0 and the Appendices. In forming such a collection of results, there will be several statements which, while interesting, will not be needed in later Lemmas, Theorems, or Examples, and can be skipped without losing any logical steps. Such statements, when proved, will be labeled “Proposition,” and otherwise labeled “Exercise,” when a short proof follows from a “Hint” or is left to the reader entirely. There are a few statements which are needed in later steps but whose proofs do not fit the basis-free theme; they are labeled “Claim,” with proofs left to the references. Adam Coffman July, 2000. v Overview The goal of these notes is to present the subject of linear algebra in a way that is both natural as its own area of mathematics, and applicable, particularly to geometry. The unifying theme is the trace operator on spaces of linear maps, and generalizations of the trace, including vector valued traces, and traces with respect to non-degenerate inner products. The emphasis is on the canonical nature of the objects and maps, and on the basis-free methods of proof. The first definition of the trace (Definition 2.3) is essentially the “conceptual” approach of Mac Lane-Birkhoff ([MB] §IX.10) and Bourbaki ([B]). This approach also is taken in disciplines using linear algebra as a tool, for example, representation theory and mathematical physics ([FH] §13.1, [Geroch] Chapter 14, [K]). Some of the subsequent formulas for the trace (Theorem 2.10, and in Section 2.4) could be used as alternate but equivalent definitions. In most cases, it is not difficult to translate the results into the usual statements about matrices and tensors, and in some cases, the proofs are more economical than choosing a basis and using matrices. In particular, no unexpected deviations from matrix theory arise. Part of the motivation for this approach is a study of vector valued Hermit- ian forms, with respect to abstractly defined complex and real structures. The conjugate linear nature of these objects necessitates careful treatment of scalar multiplication, duality of vector spaces and maps, and tensor products of vector spaces and maps ([GM], [P]). The study of Hermitian forms seems to require a pre- liminary investigation into the fundamentals of the theory of bilinear forms, which now forms the first half of these notes. The payoff from the detailed treatment of bilinear forms will be the natural way in which the Hermitian case follows, in the second half. Chapter 0 gives a brief review of elementary facts about vector spaces, as in a first college course; this should be prerequisite knowledge for most readers. Chapter 1 then sketches a review of notions of spaces of maps Hom(U, V ), tensor products U ⊗ V , and direct sums U ⊕ V , and introduces some canonical linear maps, with the notation and basic concepts which will be used in all the subsequent Chapters. Chapter 2 starts with a definition of the usual trace of a map V → V , and then states definitions for the generalized trace of maps V ⊗ U → V ⊗ W , or V → V ⊗ W , whose output is an element of Hom(U, W ), or W , respectively. Many of the theorems can be viewed as linear algebra versions of more general statements in category theory, as considered by [JSV], [Maltsiniotis], [K], [PS], [Stolz-Teichner]. Chapter 3 offers a similar basis-free approach to definitions, properties, and examples of a metric on a vector space, and the trace, or contraction, with respect to a metric. The metrics are assumed to be non-degenerate, and finite-dimensionality is a consequence. The main construction is a generalization of the well-known inner 1 2OVERVIEW product Tr(AT · B) on the space of matrices (Theorem 3.39). This could be called the “Hilbert-Schmidt” metric on Hom(U, V ), induced by arbitrary metrics on U and V ,andHom(U, V ) is shown to be isometric to U ∗ ⊗ V with the induced tensor product metric. Chapter 4 develops the W -valued case of the trace with respect to a metric. The basis-free approach is motivated in part by its usefulness in the geometry of vector bundles and structures on them, including bilinear and Hermitian forms, and almost complex structures. Important geometric applications include real vector bundles with Riemannian metrics, pseudo-Riemannian metrics (since definiteness is not assumed), or symplectic forms. The linear algebra results can be restated geo- metrically, with linear maps directly replaced by bundle morphisms, “distinguished non-zero element” by “nonvanishing section,” and in some cases, “K” by “trivial line bundle.” The plan is to proceed at an elementary pace, so that if the first few Lem- mas in Chapter 1 make sense to the reader, then nothing more advanced will be encountered after that. In particular, the relationships with differential geome- try and category theory can be ignored entirely by the uninterested reader and are mentioned here only in optional “Remarks.” It will be pointed out when the finite-dimensionality is used— for example, in the Theorems in Chapter 2 about the vector valued trace TrV ;U,W , V must be finite-dimensional, but U and W need not be. CHAPTER 0 Review of Elementary Linear Algebra 0.1. Vector spaces Definition 0.1. Given a set V ,afieldK, a binary operation + : V × V → V (addition), and a function · : K×V → V (scalar multiplication), V is a vector space means that the operations have all of the following properties: (1) Associative Law for Addition: For any u ∈ V and v ∈ V and w ∈ V , (u + v)+w = u +(v + w). (2) Existence of a Zero Element: There exists an element 0V ∈ V such that for any v ∈ V , v +0V = v. (3) Existence of an Opposite: For each v ∈ V , there exists an element of V , called −v ∈ V , such that v +(−v)=0V . (4) Associative Law for Scalar Multiplication: For any ρ, σ ∈ K and v ∈ V , (ρσ) · v = ρ · (σ · v). (5) Scalar Multiplication Identity: For any v ∈ V ,1· v = v. (6) Distributive Law: For all ρ, σ ∈ K and v ∈ V ,(ρ + σ) · v =(ρ · v)+(σ · v). (7) Distributive Law: For all ρ ∈ K and u, v ∈ V , ρ · (u + v)=(ρ · u)+(ρ · v).