DOCSLIB.ORG

Tensors: Geometry and Applications J.M. Landsberg

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Tensors: Geometry and Applications J.M. Landsberg Tensors: Geometry and Applications J.M. Landsberg Contents Preface xi 0.1. Usage xi § 0.2. Overview xii § 0.3. Clash of cultures xvii § 0.4. Further reading xviii § 0.5. Conventions, acknowledgments xix § Part 1. Motivation from applications, multilinear algebra and elementary results Chapter 1. Introduction 3 1.1. The complexity of matrix multiplication 5 § 1.2. Definitions from multilinear algebra 6 § 1.3. Tensor decomposition 11 § 1.4. P v. NP and algebraic variants 17 § 1.5. Algebraic Statistics and tensor networks 21 § 1.6. Geometry and representation theory 24 § Chapter 2. Multilinear algebra 27 2.1. Rust removal exercises 28 § 2.2. Groups and representations 30 § 2.3. Tensor products 32 § 2.4. The rank and border rank of a tensor 35 § 2.5. Examples of invariant tensors 39 § v vi Contents 2.6. Symmetric and skew-symmetric tensors 40 § 2.7. Polynomials on the space of matrices 48 § 2.8. Decomposition of V ⊗3 52 § 2.9. Appendix: Basic definitions from algebra 55 § 2.10. Appendix: Jordan and rational canonical form 57 § 2.11. Appendix: Wiring diagrams 58 § Chapter 3. Elementary results on rank and border rank 67 3.1. Ranks of tensors 68 § 3.2. Symmetric rank 70 § 3.3. Uniqueness of CP decompositions 72 § 3.4. First tests of border rank: flattenings 74 § 3.5. Symmetric border rank 76 § 3.6. Partially symmetric tensor rank and border rank 79 § 3.7. Two useful techniques for determining border rank 79 § 3.8. Strassen’s equations and variants 81 § 3.9. Equations for small secant varieties 86 § 3.10. Equations for symmetric border rank 88 § 3.11. Tensors in C2 Cb Cc 92 § ⊗ ⊗ Part 2. Geometry and Representation Theory Chapter 4. Algebraic geometry for spaces of tensors 97 4.1. Diagnostic test for those familiar with algebraic geometry 98 § 4.2. First definitions 98 § 4.3. Examples of algebraic varieties 101 § 4.4. Defining equations of Veronese re-embeddings 105 § 4.5. Grassmannians 106 § 4.6. Tangent and cotangent spaces to varieties 107 § 4.7. G-varieties and homogeneous varieties 110 § 4.8. Exercises on Jordan normal form and geometry 111 § 4.9. Further information regarding algebraic varieties 111 § Chapter 5. Secant varieties 117 5.1. Joins and secant varieties 118 § 5.2. Geometry of rank and border rank 120 § 5.3. Terracini’s lemma and first consequences 122 § Contents vii 5.4. The polynomial Waring problem 125 § 5.5. Dimensions of secant varieties of Segre Varieties 127 § 5.6. Ideas of proofs of dimensions of secant varieties of triple Segre § products 130 5.7. BRPP and conjectures of Strassen and Comon 132 § Chapter 6. Exploiting symmetry: Representation theory for spaces of tensors 137 6.1. Schur’s lemma 138 § 6.2. Finite groups 139 § 6.3. Representations of the permutation group Sd 140 § ⊗d 6.4. Decomposing V as a GL(V ) module with the aid of Sd 144 § d 6.5. Decomposing S (A An) as a G = GL(A ) § 1⊗···⊗ 1 ×···× GL(An)-module 149 6.6. Characters 151 § 6.7. The Littlewood-Richardson rule 153 § 6.8. Weights and weight spaces: a generalization of eigenvalues § and eigenspaces 157 6.9. Homogeneous varieties 164 § 6.10. Ideals of homogeneous varieties 167 § 6.11. Symmetric functions 170 § Chapter 7. Tests for border rank: Equations for secant varieties 173 7.1. Subspace varieties and multi-linear rank 174 § 7.2. Additional auxiliary varieties 177 § 7.3. Flattenings 179 § 7.4. Inheritance 184 § 7.5. Prolongation and multi-prolongation 186 § 7.6. Strassen’s equations, applications and generalizations 192 § 7.7. Equations for σ (Seg(PA PB PC)) 198 § 4 × × 7.8. Young flattenings 201 § Chapter 8. Additional varieties useful for spaces of tensors 207 8.1. Tangential varieties 208 § 8.2. Dual varieties 211 § 8.3. The Pascal determinant 214 § 8.4. Differential invariants of projective varieties 215 § 8.5. Stratifications of PV ∗ via dual varieties 219 § viii Contents 8.6. The Chow variety of zero cycles and its equations 221 § 8.7. The Fano variety of lines on a variety 225 § Chapter 9. Rank 229 9.1. Remarks on rank for arbitrary varieties 229 § 9.2. Bounds on symmetric rank 231 § 9.3. Examples of classes of polynomials and their ranks 235 § Chapter 10. Normal forms for small tensors 243 10.1. Vector spaces with a finite number of orbits 244 § 10.2. Vector spaces where the orbits can be explicitly parametrized246 § 10.3. Points in C2 Cb Cc 249 § ⊗ ⊗ 10.4. Ranks and border ranks of elements of S3C3 255 § 10.5. Tensors in C3 C3 C3 257 § ⊗ ⊗ 10.6. Normal forms for C2 S2W 258 § ⊗ 10.7. Exercises on normal forms for general points on small secant § varieties 259 10.8. Limits of secant planes 259 § 10.9. Limits for Veronese varieties 261 § 10.10. Ranks and normal forms in σ (Seg(PA PAn)) 264 § 3 1⊗···⊗ Part 3. Applications Chapter 11. The complexity of matrix multiplication 271 11.1. “Real world” issues 272 § 11.2. Failure of the border rank version of Strassen’s conjecture 273 § 11.3. Finite group approach to upper bounds 277 § 11.4. R(M , , ) 23 279 § 3 3 3 ≤ 11.5. Bl¨aser’s 5 -Theorem 279 § 2 11.6. The Brockett-Dobkin Theorem 281 § 11.7. Multiplicative complexity 283 § Chapter 12. Tensor decomposition 285 12.1. Cumulants 286 § 12.2. Blind deconvolution of DS-CMDA signals 289 § 12.3. Uniqueness results coming from algebraic geometry 295 § 12.4. Exact decomposition algorithms 297 § 12.5. Kruskal’s Theorem and its proof 301 § Contents ix Chapter 13. P v. NP 307 13.1. Introduction to complexity 308 § 13.2. Polynomials arising in complexity theory, graph theory, and § statistics 311 13.3. Definitions of VP, VNP and other algebraic complexity § classes 313 13.4. Complexity of perm and detn 318 § n 13.5. Immanants and their symmetries 325 § 13.6. Geometric Complexity Theory approach to VPws v. VNP 329 § 13.7. Other complexity classes via polynomials 336 § 13.8. Vectors of minors and homogeneous varieties 337 § 13.9. Holographic algorithms and spinors 344 § Chapter 14. Varieties of tensors in phylogenetics and quantum mechanics 353 14.1. Tensor network states 353 § 14.2. Algebraic Statistics and phylogenetics 359 § Part 4. Advanced topics Chapter 15. Overview of the proof of the Alexander-Hirschowitz theorem 369 15.1. The semi-classical cases 370 § 15.2. The Alexander-Hirschowitz idea for dealing with the § remaining cases 373 Chapter 16. Representation theory 377 16.1. Basic definitions 377 § 16.2. Casimir eigenvalues and Kostant’s theorem 381 § 16.3. Cohomology of homogeneous vector bundles 385 § 16.4. Equations and Inheritance in a more general context 389 § Chapter 17. Weyman’s method 391 17.1. Ideals and coordinate rings of projective varieties 392 § 17.2. Koszul sequences 393 § 17.3. The Kempf-Weyman method 396 § 17.4. Subspace varieties 400 § Hints and Answers to Selected Exercises 405 Bibliography 411 x Contents Index 429 Preface Tensors are ubiquitous in the sciences. One reason for their ubiquity is that they provide a useful way to organize data. Geometry is a powerful tool for extracting information from data sets, and a beautiful subject in its own right. This book has three intended uses: as a classroom textbook, a reference work for researchers, and a research manuscript. 0.1. Usage Classroom uses. Here are several possible courses one could give from this text: (1) The first part of this text is suitable for an advanced course in multi- linear algebra - it provides a solid foundation for the study of tensors and contains numerous applications, exercises, and examples. Such a course would cover Chapters 1,2,3 and parts of Chapters 4,5,6. (2) For a graduate course on the geometry of tensors not assuming al- gebraic geometry, one can cover Chapters 1,2,4,5,6,7 and 8 skipping 2.9-12, 4.6, 5.7, 6.7 (except Pieri), 7.6 and 8.6-8. (3) For a graduate course on the geometry of tensors assuming alge- braic geometry and with more emphasis on theory, one can follow the above outline only skimming Chapters 2 and 4 (but perhaps add 2.12) and add selected later topics. (4) I have also given a one semester class on the complexity of ma- trix multiplication using selected material from earlier chapters and then focusing on Chapter 11. xi xii Preface (5) Similarly I have used chapter 13 as a basis of a semester long class on P v. NP, assuming some algebraic geometry. Here Chapter 8 is important. (6) I have also given several intensive short classes on various topics. A short class for statisticians, focusing on cumulants and tensor decomposition is scheduled for the near future. Reference uses. I have compiled information on tensors in table format (e.g. regarding border rank, maximal rank, typical rank etc.) for easy reference. In particular, Chapter 3 contains most what is known on rank and border rank, stated in elementary terms. Up until now there had been no reference for even the classical results regarding tensors. (Caveat: I do not include results relying on a metric or Hermitian metric.) Research uses. I have tried to state all the results and definitions from geometry and representation theory needed to study tensors. When proofs are not included, references for them are given. The text includes the state of the art regarding ranks and border ranks of tensors, and explains for the first time many results and problems coming from outside mathematics in geometric language. For example, a very short proof of the well-known Kruskal theorem is presented, illustrating that it hinges upon a basic geomet- ric fact about point sets in projective space.
Load more

Recommended publications
  • A Some Basic Rules of Tensor Calculus
    A Some Basic Rules of Tensor Calculus The tensor calculus is a powerful tool for the description of the fundamentals in con- tinuum mechanics and the derivation of the governing equations for applied prob- lems. In general, there are two possibilities for the representation of the tensors and the tensorial equations: – the direct (symbolic) notation and – the index (component) notation The direct notation operates with scalars, vectors and tensors as physical objects defined in the three dimensional space. A vector (first rank tensor) a is considered as a directed line segment rather than a triple of numbers (coordinates). A second rank tensor A is any finite sum of ordered vector pairs A = a b + ... +c d. The scalars, vectors and tensors are handled as invariant (independent⊗ from the choice⊗ of the coordinate system) objects. This is the reason for the use of the direct notation in the modern literature of mechanics and rheology, e.g. [29, 32, 49, 123, 131, 199, 246, 313, 334] among others. The index notation deals with components or coordinates of vectors and tensors. For a selected basis, e.g. gi, i = 1, 2, 3 one can write a = aig , A = aibj + ... + cidj g g i i ⊗ j Here the Einstein’s summation convention is used: in one expression the twice re- peated indices are summed up from 1 to 3, e.g. 3 3 k k ik ik a gk ∑ a gk, A bk ∑ A bk ≡ k=1 ≡ k=1 In the above examples k is a so-called dummy index. Within the index notation the basic operations with tensors are defined with respect to their coordinates, e.
    [Show full text]
  • Tensor Algebra
    TENSOR ALGEBRA Continuum Mechanics Course (MMC) - ETSECCPB - UPC Introduction to Tensors Tensor Algebra 2 Introduction SCALAR , , ... v VECTOR vf, , ... MATRIX σε,,... ? C,... 3 Concept of Tensor A TENSOR is an algebraic entity with various components which generalizes the concepts of scalar, vector and matrix. Many physical quantities are mathematically represented as tensors. Tensors are independent of any reference system but, by need, are commonly represented in one by means of their “component matrices”. The components of a tensor will depend on the reference system chosen and will vary with it. 4 Order of a Tensor The order of a tensor is given by the number of indexes needed to specify without ambiguity a component of a tensor. a Scalar: zero dimension 3.14 1.2 v 0.3 a , a Vector: 1 dimension i 0.8 0.1 0 1.3 2nd order: 2 dimensions A, A E 02.40.5 ij rd A , A 3 order: 3 dimensions 1.3 0.5 5.8 A , A 4th order … 5 Cartesian Coordinate System Given an orthonormal basis formed by three mutually perpendicular unit vectors: eeˆˆ12,, ee ˆˆ 23 ee ˆˆ 31 Where: eeeˆˆˆ1231, 1, 1 Note that 1 if ij eeˆˆi j ij 0 if ij 6 Cylindrical Coordinate System x3 xr1 cos x(,rz , ) xr2 sin xz3 eeeˆˆˆr cosθθ 12 sin eeeˆˆˆsinθθ cos x2 12 eeˆˆz 3 x1 7 Spherical Coordinate System x3 xr1 sin cos xrxr, , 2 sin sin xr3 cos ˆˆˆˆ x2 eeeer sinθφ sin 123sin θ cos φ cos θ eeeˆˆˆ cosφφ 12sin x1 eeeeˆˆˆˆφ cosθφ sin 123cos θ cos φ sin θ 8 Indicial or (Index) Notation Tensor Algebra 9 Tensor Bases – VECTOR A vector v can be written as a unique linear combination of the three vector basis eˆ for i 1, 2, 3 .
    [Show full text]
  • Geometric Algebra Techniques for General Relativity
    Geometric Algebra Techniques for General Relativity Matthew R. Francis∗ and Arthur Kosowsky† Dept. of Physics and Astronomy, Rutgers University 136 Frelinghuysen Road, Piscataway, NJ 08854 (Dated: February 4, 2008) Geometric (Clifford) algebra provides an efficient mathematical language for describing physical problems. We formulate general relativity in this language. The resulting formalism combines the efficiency of differential forms with the straightforwardness of coordinate methods. We focus our attention on orthonormal frames and the associated connection bivector, using them to find the Schwarzschild and Kerr solutions, along with a detailed exposition of the Petrov types for the Weyl tensor. PACS numbers: 02.40.-k; 04.20.Cv Keywords: General relativity; Clifford algebras; solution techniques I. INTRODUCTION Geometric (or Clifford) algebra provides a simple and natural language for describing geometric concepts, a point which has been argued persuasively by Hestenes [1] and Lounesto [2] among many others. Geometric algebra (GA) unifies many other mathematical formalisms describing specific aspects of geometry, including complex variables, matrix algebra, projective geometry, and differential geometry. Gravitation, which is usually viewed as a geometric theory, is a natural candidate for translation into the language of geometric algebra. This has been done for some aspects of gravitational theory; notably, Hestenes and Sobczyk have shown how geometric algebra greatly simplifies certain calculations involving the curvature tensor and provides techniques for classifying the Weyl tensor [3, 4]. Lasenby, Doran, and Gull [5] have also discussed gravitation using geometric algebra via a reformulation in terms of a gauge principle. In this paper, we formulate standard general relativity in terms of geometric algebra. A comprehensive overview like the one presented here has not previously appeared in the literature, although unpublished works of Hestenes and of Doran take significant steps in this direction.
    [Show full text]
  • Tensor, Exterior and Symmetric Algebras
    Tensor, Exterior and Symmetric Algebras Daniel Murfet May 16, 2006 Throughout this note R is a commutative ring, all modules are left R-modules. If we say a ring is noncommutative, we mean it is not necessarily commutative. Unless otherwise specified, all rings are noncommutative (except for R). If A is a ring then the center of A is the set of all x ∈ A with xy = yx for all y ∈ A. Contents 1 Definitions 1 2 The Tensor Algebra 5 3 The Exterior Algebra 6 3.1 Dimension of the Exterior Powers ............................ 11 3.2 Bilinear Forms ...................................... 14 3.3 Other Properties ..................................... 18 3.3.1 The determinant formula ............................ 18 4 The Symmetric Algebra 19 1 Definitions Definition 1. A R-algebra is a ring morphism φ : R −→ A where A is a ring and the image of φ is contained in the center of A. This is equivalent to A being an R-module and a ring, with r · (ab) = (r · a)b = a(r · b), via the identification of r · 1 and φ(r). A morphism of R-algebras is a ring morphism making the appropriate diagram commute, or equivalently a ring morphism which is also an R-module morphism. In this section RnAlg will denote the category of these R-algebras. We use RAlg to denote the category of commutative R-algebras. A graded ring is a ring A together with a set of subgroups Ad, d ≥ 0 such that A = ⊕d≥0Ad as an abelian group, and st ∈ Ad+e for all s ∈ Ad, t ∈ Ae.
    [Show full text]
  • A Treatise on Quantum Clifford Algebras Contents
    A Treatise on Quantum Clifford Algebras Habilitationsschrift Dr. Bertfried Fauser arXiv:math/0202059v1 [math.QA] 7 Feb 2002 Universitat¨ Konstanz Fachbereich Physik Fach M 678 78457 Konstanz January 25, 2002 To Dorothea Ida and Rudolf Eugen Fauser BERTFRIED FAUSER —UNIVERSITY OF KONSTANZ I ABSTRACT: Quantum Clifford Algebras (QCA), i.e. Clifford Hopf gebras based on bilinear forms of arbitrary symmetry, are treated in a broad sense. Five al- ternative constructions of QCAs are exhibited. Grade free Hopf gebraic product formulas are derived for meet and join of Graßmann-Cayley algebras including co-meet and co-join for Graßmann-Cayley co-gebras which are very efficient and may be used in Robotics, left and right contractions, left and right co-contractions, Clifford and co-Clifford products, etc. The Chevalley deformation, using a Clif- ford map, arises as a special case. We discuss Hopf algebra versus Hopf gebra, the latter emerging naturally from a bi-convolution. Antipode and crossing are consequences of the product and co-product structure tensors and not subjectable to a choice. A frequently used Kuperberg lemma is revisited necessitating the def- inition of non-local products and interacting Hopf gebras which are generically non-perturbative. A ‘spinorial’ generalization of the antipode is given. The non- existence of non-trivial integrals in low-dimensional Clifford co-gebras is shown. Generalized cliffordization is discussed which is based on non-exponentially gen- erated bilinear forms in general resulting in non unital, non-associative products. Reasonable assumptions lead to bilinear forms based on 2-cocycles. Cliffordiza- tion is used to derive time- and normal-ordered generating functionals for the Schwinger-Dyson hierarchies of non-linear spinor field theory and spinor electro- dynamics.
    [Show full text]
  • 9 the Tensor Algebra
    J. Zintl: Part 3: The Tensor Product 9 THE TENSOR ALGEBRA 9 The tensor algebra 9.1 Multi-fold tensor products Up to this point, we have been considering tensor products of pairs of mod- ules M1 and M2 over a ring R. In many applications, the modules involved are free of finite ranks. In this special case, corollary ?? implies that we ∗ can identify M1 ⊗R M2 with Hom R(M1 ;M2). So for purely computational purposes, working with matrices would suffice in these cases. The theory of multilinear algebra unfolds its full strength in the natural generalization to tensor products of several R-modules M1, M2,..., Mp for some p ≥ 2. Throughout this section let (R; +; ·) always be a commutative ring with a multiplicative identity element. 2 2 9.1 Example. Consider M := R as an R-module. Let fe1; e2g ⊆ R 2 2 2 denote the standard basis of R . The standard inner product h ; i : R ×R ! R is a bilinear map. It is easy to see that the map 2 2 2 ' : R × R × R ! R (u; v; w) 7! hu; vi · hw; e1i is 3-linear. We claim that all of the information about the map ' can be recovered from the family of real numbers f'(ei; ej; ek)g1≤i;j;k≤2. Indeed, for 2 2 2 an arbitrary element (u; v; w) = ((u1; u2); (v1; v2); (w1; w2)) 2 R × R × R , we compute applying the rules for multilinear maps '(u; v; w) = '(u1e1 + u2e2; v1e1 + v2e2; w1e1 + w2e2) P = uivjwk'(ei; ej; ek): 1≤i;j;k≤2 In analogy to lemma 5.10 ??, we could think of the family fai;j;kg1≤i;j;k≤2 := f'(ei; ej; ek)g1≤i;j;k≤2 as a \3-dimensional matrix", or a 2 × 2 × 2 cube with entries in R, which represents the map ' by a suitably defined vector mul- 1 J.
    [Show full text]
  • Two-Spinors and Symmetry Operators
    Two-spinors and Symmetry Operators An Investigation into the Existence of Symmetry Operators for the Massive Dirac Equation using Spinor Techniques and Computer Algebra Tools Master’s thesis in Physics Simon Stefanus Jacobsson DEPARTMENT OF MATHEMATICAL SCIENCES CHALMERS UNIVERSITY OF TECHNOLOGY Gothenburg, Sweden 2021 www.chalmers.se Master’s thesis 2021 Two-spinors and Symmetry Operators ¦ An Investigation into the Existence of Symmetry Operators for the Massive Dirac Equation using Spinor Techniques and Computer Algebra Tools SIMON STEFANUS JACOBSSON Department of Mathematical Sciences Division of Analysis and Probability Theory Mathematical Physics Chalmers University of Technology Gothenburg, Sweden 2021 Two-spinors and Symmetry Operators An Investigation into the Existence of Symmetry Operators for the Massive Dirac Equation using Spinor Techniques and Computer Algebra Tools SIMON STEFANUS JACOBSSON © SIMON STEFANUS JACOBSSON, 2021. Supervisor: Thomas Bäckdahl, Mathematical Sciences Examiner: Simone Calogero, Mathematical Sciences Master’s Thesis 2021 Department of Mathematical Sciences Division of Analysis and Probability Theory Mathematical Physics Chalmers University of Technology SE-412 96 Gothenburg Telephone +46 31 772 1000 Typeset in LATEX Printed by Chalmers Reproservice Gothenburg, Sweden 2021 iii Two-spinors and Symmetry Operators An Investigation into the Existence of Symmetry Operators for the Massive Dirac Equation using Spinor Techniques and Computer Algebra Tools SIMON STEFANUS JACOBSSON Department of Mathematical Sciences Chalmers University of Technology Abstract This thesis employs spinor techniques to find what conditions a curved spacetime must satisfy for there to exist a second order symmetry operator for the massive Dirac equation. Conditions are of the form of the existence of a set of Killing spinors satisfying some set of covariant differential equations.
    [Show full text]
  • On Electromagnetic Duality
    On Electromagnetic Duality Thomas B. Mieling Faculty of Physics, University of Vienna Boltzmanngasse 5, 1090 Vienna, Austria (Dated: November 14, 2018) CONTENTS III. DUAL TENSORS I. Introduction 1 This section introduces the notion of dual ten- sors, whose proper treatment requires some famil- II. Conventions 1 iarity with exterior algebra. Readers unfamiliar with this topic may skip section III A, since sec- III. Dual Tensors 1 tion III B suces for the calculations in the rest of A. The Hodge Dual 1 the paper, except parts of the appendix. B. The Complex Dual of Two-Forms 2 IV. The Free Maxwell-Field 2 A. The Hodge Dual V. General Duality Transformations 3 For an oriented vector space V of dimension VI. Coupled Maxwell-Fields 3 n with a metric tensor, the Hodge star opera- tor provides an isomorphism between the k-vectors VII. Applications 4 n − k-vectors. In this section, we discuss the gen- eral denition in Euclidean and Lorentzian vector VIII. Conclusion 5 spaces and give formulae for concrete calculations in (pseudo-)orthonormal frames. A. Notes on Exterior Algebra 6 The metric tensor g on V induces a metric tensor on ^kV (denoted by the same symbol) by bilinear B. Collection of Proofs 7 extension of the map References 8 g(v1 ^ · · · ^ vk; w1 ^ · · · ^ wk) = det(g(vi; wj)): If the metric on V is non-degenerate, so is g on I. INTRODUCTION ^kV . Let w be a k-vector. For every n − k-vector In abelian gauge theories whose action does not v the exterior product v ^ w is proportional to depend on the gauge elds themselves, but only the unit n-vector Ω, since ^nV is one-dimensional.
    [Show full text]
  • Tensor Algebra and Tensor Calculus
    Appendix A Tensor Algebra and Tensor Calculus We collect here some notions of geometry and analysis used in the previous chapters. This appendix is no more than a brief summary containing a few defi- nitions and statements of results without proof. The conscientious reader unfamiliar with the topics collected here should take from these pages some directions for further reading in appropriate textbooks. A.1 Vector Spaces Definition 1. AsetV is called a vector space (or a linear space) over a field F if it is endowed with two operations, CWV V ! V and ıWV F ! V, called respectively addition and multiplication, such that – addition is commutative, i.e., v C w D w C v for every pair v;w 2 V, – addition is associative, i.e., .v Cw/Cz D v C.wCz/ for every triple v;w; z 2 V, – addition admits a unique neutral element 0 2 V such that v C 0 D v;8v 2 V, – with ˛v denoting the product ˛ ıv, for every ˛; ˇ 2 F and v 2 V, multiplication is associative, i.e., ˛.ˇv/ D .˛ˇ/v, where ˛ˇ is the product in F, – multiplication admits a unique neutral element 1 2 F such that 1v D v;8v 2 V, – multiplication is distributive with respect to addition, i.e., ˛.v C w/ D ˛v C ˛w for every pair v;w 2 V, – multiplication is distributive with respect to addition in F, i.e., for every pair ˛; ˇ 2 F and every v 2 V, .˛ C ˇ/v D ˛v C ˇv: We do not recall the abstract definition of a field in the algebraic sense, because we consider here only the special case of F equal to the set of real numbers R.
    [Show full text]
  • A Basic Operations of Tensor Algebra
    A Basic Operations of Tensor Algebra The tensor calculus is a powerful tool for the description of the fundamentals in con- tinuum mechanics and the derivation of the governing equations for applied prob- lems. In general, there are two possibilities for the representation of the tensors and the tensorial equations: – the direct (symbolic, coordinate-free) notation and – the index (component) notation The direct notation operates with scalars, vectors and tensors as physical objects defined in the three-dimensional space (in this book we are limit ourselves to this case). A vector (first rank tensor) a is considered as a directed line segment rather than a triple of numbers (coordinates). A second rank tensor A is any finite sum of ordered vector pairs A = a ⊗ b + ...+ c ⊗ d. The scalars, vectors and tensors are handled as invariant (independent from the choice of the coordinate system) quantities. This is the reason for the use of the direct notation in the modern literature of mechanics and rheology, e.g. [32, 36, 53, 126, 134, 205, 253, 321, 343] among others. The basics of the direct tensor calculus are given in the classical textbooks of Wilson (founded upon the lecture notes of Gibbs) [331] and Lagally [183]. The index notation deals with components or coordinates of vectors and tensors. For a selected basis, e.g. gi, i = 1, 2, 3 one can write i i j i j ⊗ a = a gi, A = a b + ...+ c d gi gj Here the Einstein’s summation convention is used: in one expression the twice re- peated indices are summed up from 1 to 3, e.g.
    [Show full text]
  • Geometric Methods in Mathematical Physics I: Multi-Linear Algebra, Tensors, Spinors and Special Relativity
    Geometric Methods in Mathematical Physics I: Multi-Linear Algebra, Tensors, Spinors and Special Relativity by Valter Moretti www.science.unitn.it/∼moretti/home.html Department of Mathematics, University of Trento Italy Lecture notes authored by Valter Moretti and freely downloadable from the web page http://www.science.unitn.it/∼moretti/dispense.html and are licensed under a Creative Commons Attribuzione-Non commerciale-Non opere derivate 2.5 Italia License. None is authorized to sell them 1 Contents 1 Introduction and some useful notions and results 5 2 Multi-linear Mappings and Tensors 8 2.1 Dual space and conjugate space, pairing, adjoint operator . 8 2.2 Multi linearity: tensor product, tensors, universality theorem . 14 2.2.1 Tensors as multi linear maps . 14 2.2.2 Universality theorem and its applications . 22 2.2.3 Abstract definition of the tensor product of vector spaces . 26 2.2.4 Hilbert tensor product of Hilbert spaces . 28 3 Tensor algebra, abstract index notation and some applications 34 3.1 Tensor algebra generated by a vector space . 34 3.2 The abstract index notation and rules to handle tensors . 37 3.2.1 Canonical bases and abstract index notation . 37 3.2.2 Rules to compose, decompose, produce tensors form tensors . 39 3.2.3 Linear transformations of tensors are tensors too . 41 3.3 Physical invariance of the form of laws and tensors . 43 3.4 Tensors on Affine and Euclidean spaces . 43 3.4.1 Tensor spaces, Cartesian tensors . 45 3.4.2 Applied tensors . 46 3.4.3 The natural isomorphism between V and V ∗ for Cartesian vectors in Eu- clidean spaces .
    [Show full text]
  • 5. SPINORS 5.1. Prologue. 5.2. Clifford Algebras and Their
    5. SPINORS 5.1. Prologue. 5.2. Clifford algebras and their representations. 5.3. Spin groups and spin representations. 5.4. Reality of spin modules. 5.5. Pairings and morphisms. 5.6. Image of the real spin group in the complex spin module. 5.7. Appendix: Some properties of the orthogonal groups. 5.1. Prologue. E. Cartan classified simple Lie algebras over C in his thesis in 1894, a classification that is nowadays done through the (Dynkin) diagrams. In 1913 he classified the irreducible finite dimensional representations of these algebras1. For any simple Lie algebra g Cartan's construction yields an irreducible representation canonically associated to each node of its diagram. These are the so-called fun- damental representations in terms of which all irreducible representations of g can be constructed using and subrepresentations. Indeed, if πj(1 j `) are the ⊗ ≤ ≤ fundamental representations and mj are integers 0, and if vj is the highest vector ≥ of πj, then the subrepresentation of m1 m` π = π⊗ ::: π⊗ 1 ⊗ ⊗ ` generated by m1 m` v = v⊗ ::: v⊗ 1 ⊗ ⊗ ` is irreducible with highest vector v, and every irreducible module is obtained in this manner uniquely. As is well-known, Harish-Chandra and Chevalley (independently) developed around 1950 a general method for obtaining the irreducible representa- tions without relying on case by case considerations as Cartan did. `+1 j If g = sl(` + 1) and V = C , then the fundamental module πj is Λ (V ), and all irreducible modules can be obtained by decomposing the tensor algebra over the 1 defining representation V . Similarly, for the symplectic Lie algebras, the decomposi- tion of the tensors over the defining representation gives all the irreducible modules.
    [Show full text]