Tensors: Geometry and Applications J.M. Landsberg - Texas A&M ...

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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 C 2 ⊗C b ⊗C c 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 §6.4. Decomposing V ⊗d as a GL(V ) module with the aid of Sd 144 §6.5. Decomposing S d (A1⊗ · · · ⊗ An) as a G = GL(A1) × · · · × 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 σ4(Seg(PA × PB × PC)) 198 §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
Page 1: Tensors: Geometry and Applications
Page 6 and 7: viii Contents §8.6. The Chow varie
Page 8 and 9: x Contents Index 429
Page 10 and 11: xii Preface (5) Similarly I have us
Page 12 and 13: xiv Preface This chapter may be dif
Page 14 and 15: xvi Preface Chapter 12: Tensor deco
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Page 18 and 19: xx Preface 0.5.3. Acknowledgments.
Page 21 and 22: Introduction Chapter 1 The workhors
Page 23 and 24: 1.1. The complexity of matrix multi
Page 25 and 26: 1.2. Definitions from multilinear a
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Page 29 and 30: 1.3. Tensor decomposition 11 In pri
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Page 33 and 34: 1.3. Tensor decomposition 15 source
Page 35 and 36: 1.4. P v. NP and algebraic variants
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Page 39 and 40: 1.5. Algebraic Statistics and tenso
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40 2. Multilinear algebra the actio
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42 2. Multilinear algebra where Sd
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44 2. Multilinear algebra There is
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50 2. Multilinear algebra which is
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60 2. Multilinear algebra Input: el
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62 2. Multilinear algebra V ⊗V
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64 2. Multilinear algebra V ⊗V
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