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

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xx Preface 0.5.3. Acknowledgments. This project started as a collaboration with Jason Morton, who contributed significantly to the writing and editing of chapters 2,4,5 and 6. The book has greatly benefitted from his input. The first draft of this book arose out of a series of lectures B. Sturmfels invited me to give for his working group at UC Berkeley in spring 2006. I then gave a graduate class at Texas A&M University in fall 2007 on the complexity of matrix multiplication and a class on complexity theory in spring 2009. J. Morton and I gave lectures from the notes at a summer graduate workshop at MSRI in July 2008, as well as several lectures at a follow-up research workshop at AIM in July 2008. I also gave a GNSAGA lecture series in Florence, Italy in June 2009 on secant varieties, a short course on the geometry of tensors June 2010 in Nordfjordeid, Norway, and a one semester graduate class on the same subject at Texas A&M, Fall 2010. It is a pleasure to thank the students in these classes as well as my hosts B. Sturmfels, MSRI, AIM, G. Ottaviani and K. Ranestad. Much of the material in this book comes from joint work with L. Manivel, G. Ottaviani, and J. Weyman. It is a pleasure to thank these three collaborators for significant help at each step along the way. Other material comes from joint work with the (at the time) post-docs J. Buczyński, J. Morton, S. Norine, and Z. Teitler, who have also significantly helped with the book. I also thank my students and post-docs A. Boralevi, L. Nguyen, L. Oeding, Y. Qi, D. The, M. Yang, and K. Ye for their useful comments and questions, as well as R. Bardeli, M. Bläser, P. Bürgisser, E. Briand, J-Y Cai, P. Comon, L. De Lathauwer, M. Dillon, M. Eastwood, J. von zur Gathen, J. Grochow, C. Hammond, L-H Lim, and G. Paouris for useful comments and patiently answering my questions. It would not have been possible to write this book without such wonderful help. I was very fortunate to have careful and thoughtful anonymous referees who made numerous outstanding suggestions for improving the manuscript for which I am truly grateful. Finally I thank my family for their help, support and understanding.
Part 1 Motivation from applications, multilinear algebra and elementary results
Page 1: Tensors: Geometry and Applications
Page 4 and 5: vi Contents §2.6. Symmetric and sk
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
Page 16 and 17: xviii Preface this and similar inst
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
Page 27 and 28: 1.2. Definitions from multilinear a
Page 29 and 30: 1.3. Tensor decomposition 11 In pri
Page 31 and 32: 1.3. Tensor decomposition 13 them t
Page 33 and 34: 1.3. Tensor decomposition 15 source
Page 35 and 36: 1.4. P v. NP and algebraic variants
Page 37 and 38: 1.4. P v. NP and algebraic variants
Page 39 and 40: 1.5. Algebraic Statistics and tenso
Page 41 and 42: 1.5. Algebraic Statistics and tenso
Page 43: 1.6. Geometry and representation th
Page 46 and 47: 28 2. Multilinear algebra I work ov
Page 48 and 49: 30 2. Multilinear algebra symmetric
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Page 52 and 53: 34 2. Multilinear algebra The numbe
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Page 56 and 57: 38 2. Multilinear algebra point for
Page 58 and 59: 40 2. Multilinear algebra the actio
Page 60 and 61: 42 2. Multilinear algebra where Sd
Page 62 and 63: 44 2. Multilinear algebra There is
Page 64 and 65: 46 2. Multilinear algebra (7) Show
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50 2. Multilinear algebra which is
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52 2. Multilinear algebra (2.7.1) (
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54 2. Multilinear algebra occur wit
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56 2. Multilinear algebra Definitio
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58 2. Multilinear algebra where ψ1
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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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