from vectors to tensors - j. ruiz-tolosa, e. castillo (springer, 2005) ww

Universitext Juan Ramon Ruiz-Tolosa Enrique Castillo From Vectors to Tensors Springer Authors Juan Ramon Ruiz-Tolosa Enrique Castillo Universidad Cantabria Depto. Matematica Aplicada Avenida de los Castros 39005 Santander, Spain castie@unican,es Library of Congress Control Number: 20041114044 Mathematics Subject Classification (2000): 15A60,15A72 ISBN3-540-22887-X Springer Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9,1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable for prosecution under the German Copyright Law. Springer-Verlag is a part of Springer Science+Business Media springeronline.com © Springer-Verlag Berlin Heidelberg 2005 Printed in Germany The use of designations, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover Design: Erich Kirchner, Heidelberg Printed on acid-free paper 41/3142XT 543210 To the memory of Bernhard Riemann and Albert Einstein Preface It is true that there exist many books dedicated to linear algebra and some- what fewer to multilinear algebra, written in several languages, and perhaps one can think that no more books are needed. However, it is also true that in algebra many new results are continuously appearing, different points of view can be used to see the mathematical objects and their associated structures, and different orientations can be selected to present the material, and all of them deserve publication. Under the leadership of Juan Ramon Ruiz-Tolosa, Professor of multilin- ear algebra, and the collaboration of Enrique Castillo, Professor of applied mathematics, both teaching at an engineering school in Santander, a tensor textbook has been born, written from a practical point of view and free from the esoteric language typical of treatises written by algebraists, who are not interested in descending to numerical details. The balance between follow- ing this line and keeping the rigor of classical theoretical treatises has been maintained throughout this book. The book assumes a certain knowledge of linear algebra, and is intended as a textbook for graduate and postgraduate students and also as a consultation book. It is addressed to mathematicians, physicists, engineers, and applied scientists with a practical orientation who are looking for powerful tensor tools to solve their problems. The book covers an existing chasm between the classic theory of tensors and the possibility of solving tensor problems with a computer. In fact, the computational algebra is formulated in matrix form to facilitate its implemen- tation on computers. The book includes 197 examples and end-of-chapter exercises, which makes it specially suitable as a textbook for tensor courses. This material combines classic matrix techniques together with novel methods and in many cases the questions and problems are solved using different methods. They confirm the applied orientation of the book. A computer package, written in Mathematica, accompanies this book, available on: http://personales.unican.es/castie/tensors. In it, most of the novel methods developed in the book have been implemented. We note that existing general computer software packages (Mathematica, Mathlab, etc.) for tensors are very poor, up to the point that some problems cannot be dealt VIII Preface with using computers because of the lack of computer programs to perform these operations. The main contributions of the book are: 1. The book employs a new technique that permits one to extend (stretch) the tensors, as one-column matrices, solve on these matrices the desired problems, and recover the initial format of the tensor (condensation). This technique, applied in all chapters, is described and used to solve matrix equations in Chapter 1. 2. An important criterion is established in Chapter 2 for all the components of a tensor to have a given ordering, by the definition of a unique canonical tensor basis. This permits the mentioned technique to be applied. 3. In Chapter 3, factors are illustrated that have led to an important con- fusion in tensor books due to inadequate notation of tensors or tensor operations. 4. In addition to dealing with the classical topics of tensor books, new tensor concepts are introduced, such as the rotation of tensors, the transposer tensor, the eigentensors, and the permutation tensor structure, in Chapter 5. 5. A very detailed study of generalized Kronecker deltas is presented in Chap- ter 8. 6. Chapter 10 is devoted to mixed exterior algebras, analyzing the problem of change-of-basis and the exterior product of this kind of tensors. 7. In Chapter 11 the rules for the "Euclidean contraction" are given in detail. This chapter ends by introducing the geometric concepts to tensors. 8. The orientation and polar tensors in Euclidean spaces are dealt with in Chapter 12. 9. In Chapter 13 the Gram matrices G(r) are established to connect exterior tensors. 10. Chapter 14 is devoted to Euclidean tensors in E^(R), affine geometric tensors (homographies), and some important tensors in physics and me- chanics, such as the stress and strain tensors, the elastic tensor and the inertial moment tensor. It is shown how tensors allow one to solve very interesting practical problems. In summary, the book is not a standard book on tensors because of its orientation, the many novel contributions included in it, the careful notation and the stretching-condensing techniques used for most of the transformations used in the book. We hope that our readers enjoy reading this book, discover a new world, and acquire stimulating ideas for their applications and new contributions and research. The authors want to thank an anonimous French referee for the careful reading of the initial manuscript, and to Jeffrey Boys for the copyediting of the final manuscript. Santander, Juan Ramon Rmz-Tolosa September 30, 2004 Enrique Castillo Contents Part I Basic Tensor Algebra Tensor Spaces 3 1.1 Introduction 3 1.2 Dual or reciprocal coordinate frames in affine Euclidean spaces 3 1.3 Different types of matrix products 8 1.3.1 Definitions 8 1.3.2 Properties concerning general matrices 10 1.3.3 Properties concerning square matrices 11 1.3.4 Properties concerning eigenvalues and eigenvectors 12 1.3.5 Properties concerning the Schur product 13 1.3.6 Extension and condensation of matrices 13 1.3.7 Some important matrix equations 17 1.4 Special tensors 26 1.5 Exercises . 30 Introduction to Tensors 33 2.1 Introduction 33 2.2 The triple tensor product linear space 33 2.3 Einstein's summation convention 36 2.4 Tensor analytical representation 37 2.5 Tensor product axiomatic properties 38 2.6 Generalization 40 2.7 Illustrative examples , 41 2.8 Exercises 46 Homogeneous Tensors 47 3.1 Introduction 47 3.2 The concept of homogeneous tensors 47 3.3 General rules about tensor notation 48 3.4 The tensor product of tensors 50 X Contents 3.5 Einstein's contraction of the tensor product 54 3.6 Matrix representation of tensors 56 3.6.1 First-order tensors 56 3.6.2 Second-order tensors 57 3.7 Exercises 61 4 Change-of-basis in Tensor Spaces 65 4.1 Introduction 65 4.2 Change of basis in a third-order tensor product space 65 4.3 Matrix representation of a change-of-basis in tensor spaces 67 4.4 General criteria for tensor character 69 4.5 Extension to homogeneous tensors 72 4.6 Matrix operation rules for tensor expressions 74 4.6.1 Second-order tensors (matrices) 74 4.6.2 Third-order tensors 77 4.6.3 Fourth-order tensors 78 4.7 Change-of-basis invariant tensors: Isotropic tensors 80 4.8 Main isotropic tensors 80 4.8.1 The null tensor 80 4.8.2 Zero-order tensor (scalar invariant) 80 4.8.3 Kronecker's delta 80 4.9 Exercises 106 5 Homogeneous Tensor Algebra: Tensor Homomorphisms Ill 5.1 Introduction Ill 5.2 Main theorem on tensor contraction Ill 5.3 The contracted tensor product and tensor homomorphisms 113 5.4 Tensor product applications 119 5.4.1 Common simply contracted tensor products 119 5.4.2 Multiply contracted tensor products 120 5.4.3 Scalar and inner tensor products 120 5.5 Criteria for tensor character based on contraction 122 5.6 The contracted tensor product in the reverse sense: The quotient law 124 5.7 Matrix representation of permutation homomorphisms 127 5.7.1 Permutation matrix tensor product types in K"^ . . . 127 5.7.2 Linear span of precedent types 129 5.7.3 The isomers of a tensor 137 5.8 Matrices associated with simply contraction homomorphisms . 141 5.8.1 Mixed tensors of second order (r = 2): Matrices 141 5.8.2 Mixed tensors of third order (r = 3) 141 5.8.3 Mixed tensors of fourth order (r = 4) 142 5.8.4 Mixed tensors of fifth order (r = 5) 143 5.9 Matrices associated with doubly contracted homomorphisms . 144 5.9.1 Mixed tensors of fourth order (r = 4) 144 Contents XI 5.9.2 Mixed tensors of fifth order (r = 5) 145 5.10 Eigentensors 159 5.11 Generahzed multihnear mappings 165 5.11.1 Theorems of simihtude with tensor mappings 167 5.11.2 Tensor mapping types 168 5.11.3 Direct n-dimensional tensor endomorphisms 169 5.12 Exercises 183 Part II Special Tensors 6 Symmetric Homogeneous Tensors: Tensor Algebras 189 6.1 Introduction 189 6.2 Symmetric systems of scalar components 189 6.2.1 Symmetric systems with respect to an index subset 190 6.2.2 Symmetric systems. Total symmetry 190 6.3 Strict components of a symmetric system 191 6.3.1 Number of strict components of a symmetric system with respect to an index subset 191 6.3.2 Number of strict components of a symmetric system . 192 6.4 Tensors with symmetries: Tensors with branched symmetry, symmetric tensors 193 6.4.1 Generation of symmetric tensors 194 6.4.2 Intrinsic character of tensor symmetry: Fundamental theorem of tensors with symmetry 197 6.4.3 Symmetric tensor spaces and subspaces. Strict components associated with subspaces 204 6.5 Symmetric tensors under the tensor algebra perspective 206 6.5.1 Symmetrized tensor associated with an arbitrary pure tensor 210 6.5.2 Extension of the symmetrized tensor associated with a mixed tensor 210 6.6 Symmetric tensor algebras: The (8)5 product 212 6.7 Illustrative examples 214 6.8 Exercises 220 7 Anti-symmetric Homogeneous Tensors, Tensor and Inner Product Algebras 225 7.1 Introduction 225 7.2 Anti-symmetric systems of scalar components 225 7.2.1 Anti-symmetric systems with respect to an index subset 226 7.2.2 Anti-symmetric systems. Total anti-symmetry 228 7.3 Strict components of an anti-symmetric system and with respect to an index subset 228 XII Contents 7.3.1 Number of strict components of an anti-symmetric system with respect to an index subset 229 7.3.2 Number of strict components of an anti-symmetric system 229 7.4 Tensors with anti-symmetries: Tensors with branched anti-symmetry; anti-symmetric tensors 230 7.4.1 Generation of anti-symmetric tensors 232 7.4.2 Intrinsic character of tensor anti-symmetry: Fundamental theorem of tensors with anti-symmetry . 236 7.4.3 Anti-symmetric tensor spaces and subspaces. Vector subspaces associated with strict components 243 7.5 Anti-symmetric tensors from the tensor algebra perspective . . 246 7.5.1 Anti-symmetrized tensor associated with an arbitrary pure tensor 249 7.5.2 Extension of the anti-symmetrized tensor concept associated with a mixed tensor 249 7.6 Anti-symmetric tensor algebras: The ^H product 252 7.7 Illustrative examples 253 7.8 Exercises 265 8 Pseudotensors; Modular, Relative or Weighted Tensors 269 8.1 Introduction 269 8.2 Previous concepts of modular tensor establishment 269 8.2.1 Relative modulus of a change-of-basis 269 8.2.2 Oriented vector space 270 8.2.3 Weight tensor 270 8.3 Axiomatic properties for the modular tensor concept 270 8.4 Modular tensor characteristics 271 8.4.1 Equality of modular tensors 272 8.4.2 Classification and special denominations 272 8.5 Remarks on modular tensor operations: Consequences 272 8.5.1 Tensor addition 272 8.5.2 Multiplication by a scalar 274 8.5.3 Tensor product 275 8.5.4 Tensor contraction 276 8.5.5 Contracted tensor products 276 8.5.6 The quotient law. New criteria for modular tensor character 277 8.6 Modular symmetry and anti-symmetry 280 8.7 Main modular tensors 291 8.7.1 e systems, permutation systems or Levi-Civita tensor systems 291 8.7.2 Generalized Kronecker deltas: Definition 293 8.7.3 Dual or polar tensors: Definition 301 8.8 Exercises 310 [...]... 1 2-1 / C2I C22 C23 Q, C31 \C33/ (1.44) where Cij are the elements of matrix C and the block representation has been used for illustrating the relation of the new matrix to matrices A and B Whence /-4 - 1-1 -1 - 2-1 -1 - 2-1 4 0 0 0 4 0 0 0 4 - 4 0 0\ 0-4 0 ^ 0 0-4 /Cll\ ' C12 ^ Cl3 C2I C22 C23 3 0 0 0 3 0 0 0 3 \ -2 - 1-1 - 1 0-1 -1 -2 1 4 0 0 0 4 0 0 0 4 2 0 0 0 2 0 0 0 2 - 1 0 0 0-1 0 0 0-1 1-1 -1 -1 3-1 i -. .. of eigenvectors of t h e matrix A={G-^X*YGY = {X*)\G-'^YGY = {X*YG-'^GY = {X*flY, (1.10) where once more we observe t h a t cova-contravariant d a t a imply a unit connection matrix / 4 Finally, if one has cova-covariant data, t h a t is, V{X*) and 1^(1^*), the result will b e - X^GY - (G-iX*)*G(G-^y*) = {XyG-^GG-^Y'' (1.11) = (X*)*G-^y*, which discovers... 8pi - p2 + 5p3 8Pi + P2 - 13P3 \ C 1 7 p i + p 2 + 5p3 7 p i + p 2 + P3 V 9pi P2 (1.46) 7pi 4-1 0p3 9P3, ) where pi,P2 and p3 are arbitrary real constants Its determinant is \C\ = (9p3 - P2)(90p? - 9pip2 -PI- 9pip3 - 9p2P3 - 18pi), and thus, the most general change-of-basis matrix that transforms matrix A into matrix 5 , by the similarity transformation, C~^AC = ß , is that given by (1.46) subject to. .. 2^32 ^33 i7, yii yi2 y2i y22 I ysi \ y32 / and solving t h e resulting homogeneous system a n d condensing X and Y one finally gets X = -pS - 2p4 - 4p7 -P4 - 4p6 - 2 p 3 - P4 - 4/95 Pi + 2p2 - 2/93 - 4p4 + P7 P 2- 2p4 + P6 2/9i + /92 " 4/93 - 2p4 + Pö 3/96 3P5 3p7 y = -4 ^2 + 6/94 3/94 P2 D where /9i, p2r • • • 5 P? ^ire arbitrary real constants ... cited basis) transforms the vectors in the affine linear space into vectors of t h e same space In this situation, one performs a change-of-basis in E'^CSl) (with given associated matrix C) We are interested in finding the m a t r i x M associated with t h e linear operator, such t h a t taking vectors in contravariant coordinates of t h e initial basis returns the transformed vectors in "covariant coordinates"... = ß, which is of the form (1.41); thus, after stretching the matrix C, we get (see Equation (1.42)): 4 0 0 - 4 0 0\ 0 4 0 0 - 4 0^ 0 0 4 0 0-4 /-2 0 0 ^ 0-2 0 0 0-2 0 0 0 0 0 0 0 0 0 3 0 0 0 3 0 0 0 3 M • c = (^ (g) Js - I3 (g) 5*) • c =: 2 0 0 -1 0 0 , 0 2 0 0-1 0 \ 0 0 2 0 0-1 /21 1 ' 10 1 1 2-1 00 0 00 0 00 0 00 0 00 0 00 0 /Cll\ ' C12 ^ Cl3 4 0 0 0 4 0 0 0 4 3 0 0 0 3 0, 0 0 3/ 00 ^ 00 0 \ 00 0 00... Cartesian tensors 11.10 Euclidean and pseudo-Euclidean tensor algebra 11.10.1 Euclidean tensor equality 11.10.2 Addition and external product of Euclidean (pseudo-Euclidean) tensors 11.10.3 Tensor product of Euclidean (pseudo-Euclidean) tensors 11.10.4 Euclidean (pseudo-Euclidean) tensor contraction 11.10.5 Contracted tensor product of Euclidean or pseudo-Euclidean tensors 11.10.6 Euclidean contraction of tensors. .. f i = ( l + [ ^ J ) + [(j-l)-p Plij ~- , 0 otherwise • 7-1 p m ; z,j = l , , m p , (1.34) where [xj is the integer part of x, and ^ Ruiz-Tolosa and Castillo [48] have generalized these equations to tensor equations 1 Tensor Spaces 18 P2{n,q) =Pnq,nq = [P2ij where P2ij lifj= i-l 1+ i,j = 1, I 0 otherwise ,nq, (1.35) (1.36) which shows that Remark 1.3 It is interesting to check that the matrices . condense tensors very frequently, i .e. , we represent tensors as vectors to take full advantage of vector theory and tools, and then we recover their initial tensor representation, we present the. the eigentensors, and the permutation tensor structure, in Chapter 5. 5. A very detailed study of generalized Kronecker deltas is presented in Chap- ter 8. 6. Chapter 10 is devoted to mixed. components of exterior vectors. Multivectors. 318 9.2 Exterior product of r vectors: Decomposable multivectors 319 9.2.1 Properties of exterior products of order r: Decomposable multivectors