Tensors and Hypermatrices

Handbook of Linear Algebra, Second Edition

Volumn , Issue , 2013, Pages

  Lim, Lek Heng ^a  

^a UNIVERSITY OF CHICAGO   (United States)

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EID: 85131471036     PISSN: None     EISSN: None     Source Type: Book    
DOI: None     Document Type: Chapter

References (45)

1. W.A. Adkins and S.H.Weintraub. Algebra: An Approach Via Module Theory. Springer-Verlag, New York, 1992.
2. R.J. Baxter. Solvable eight-vertex model on an arbitrary planar lattice. Philos. Trans. Roy. Soc. London Ser. A, 289 (1359): 315-346, 1978.
3. G.M. Bergman, Ranks of tensors and change of base field, J. Algebra, 11: 613-621, 1969.
4. D. Bini, G. Lotti, and F. Romani. Approximate solutions for the bilinear form computational problem. SIAM J. Comput., 9 (4): 692-697, 1980.
5. S.F. Borg. Matrix-Tensor Methods in Continuum Mechanics, 2nd ed. World Scientific, Singapore, 1990.
6. N. Bourbaki. Algebra I: Chapters 1-3. Springer-Verlag, Berlin, 1998.
7. J.-L. Brylinski. Algebraic measures of entanglement. In R. K. Brylinski and G. Chen (Eds.), Mathematics of Quantum Computation, pp. 3-23, CRC Press, Boca Raton, FL, 2002.
8. P. Bürgisser, M. Clausen, and M.A. Shokrollahi. Algebraic Complexity Theory, Springer-Verlag, Berlin, 1996.
9. T.K. Caughey, M.E.J. O’Kelly. Classical normal modes in damped linear dynamic systems. ASME J. Applied Mechanics, 32: 583-588, 1965.
10. A. Cayley. On the theory of linear transformations. Cambridge Math. J., 4: 193-209, 1854.
11. A. Cayley. On the theory of determinants. Trans. Cambridge Philos. Soc., 8, no. 7: 75-88, 1849.
12. J.F. Cornwell. Group Theory in Physics, Vol. II. Academic Press, London, 1984.
13. A. Defant and K. Floret. Tensor Norms and Operator Ideals. North-Holland, Amsterdam, 1993.
14. H. Derksen. Sharpness of Kruskal’s theorem. Linear Algebra Appl., 438(2): 708-712, 2013.
15. H. Derksen. Personal communication, 2013.
16. D.L. Donoho and M. Elad. Optimally sparse representation in general (nonorthogonal) dictionaries via `1 minimization. Proc. Nat. Acad. Sci., 100(5): 2197-2202, 2003.
17. W. Dür, G. Vidal, and J.I. Cirac. Three qubits can be entangled in two inequivalent ways. Phys. Rev. A, 62(6), 062314, 12 pp., 2000.
18. I.M. Gelfand, M.M. Kapranov, and A.V. Zelevinsky. Discriminants, Resultants, and Multidimensional Determinants. Birkhäuser Publishing, Boston, MA, 1994.
19. I.M. Gelfand, M.M. Kapranov, and A.V. Zelevinsky. Hyperdeterminants. Adv. Math., 96(2): 226-263, 1992.
20. D.M. Greenberger, M.A. Horne, and A. Zeilinger. Going beyond Bell’s theorem, In M. Kafatos (Ed.), Bell’s Theorem, Quantum Theory and Conceptions of the Universe, pp. 69-72, Kluwer, Dordrecht, Netherlands, 1989.
21. W. Greub. Multilinear Algebra, 2nd ed. Springer-Verlag, New York, 1978.
22. D. Grier, P. Huggins, B. Sturmfels, and J. Yu. The hyperdeterminant and triangulations of the 4-cube. Math. Comp., 77(263): 1653-1679, 2008.
23. P. Halmos. I Want to Be a Mathematician::: An Automathography. Springer-Verlag, New York, 1985.
24. S. Hill and W.K. Wootters. Entanglement of a pair of quantum bits. Phys. Rev. Lett., 78(26): 5022-5025, 1997.
25. F.L. Hitchcock. The expression of a tensor or a polyadic as a sum of products. J. Math. Phys., 6(1): 164-189, 1927.
26. F.L. Hitchcock. Multiple invariants and generalized rank of a p-way matrix or tensor. J. Math. Phys., 7(1): 39-79, 1927.
27. J.E. Hopcroft and L.R. Kerr. On minimizing the number of multiplications necessary for matrix multiplication. SIAM J. Appl. Math., 20(1): 30-36, 1971.
28. D. Knuth. The Art of Computer Programming 2: Seminumerical Algorithms, 3rd ed. Addison-Wesley, Reading, MA, 1998.
29. A.I. Kostrikin and Y.I. Manin. Linear Algebra and Geometry. Gordon and Breach, Amsterdam, 1997.
30. J.B. Kruskal. Three-way arrays: rank and uniqueness of trilinear decompositions, with application to arithmetic complexity and statistics. Lin. Alg. Appl., 18(2): 95-138, 1977.
31. J.M. Landsberg. The border rank of the multiplication of 2 × 2 matrices is seven. J. Amer. Math. Soc., 19(2): 447-459, 2006.
32. J.M. Landsberg. Tensors: Geometry and Applications. AMS, Providence, RI, 2012.
33. S. Lang. Algebra, Rev. 3rd ed. Springer-Verlag, New York, 2002.
34. L.-H. Lim. Tensors and hypermatrices. Preprint, http://www.stat.uchicago.edu/~lekheng/work/tensors.pdf.
35. M. Marcus. Finite Dimensional Multilinear Algebra, Parts I and II. Marcel Dekker, New York, 1973 and 1975.
36. A. Miyake and M. Wadati. Multipartite entanglement and hyperdeterminant. Quantum Inf. Comput., 2: 540-555, 2002.
37. D.G. Northcott. Multilinear Algebra. Cambridge University Press, Cambridge, 1984.
38. J.G. Oxley. Matroid Theory, 2nd ed. Oxford University Press, Oxford, 2011.
39. N.D. Sidiropoulos and R. Bro. On the uniqueness of multilinear decomposition of N-way arrays. J. Chemometrics, 14(3): 229-239, 2000.
40. V. Strassen. Gaussian elimination is not optimal. Numer. Math., 13(4): 354-356, 1969.
41. V. Strassen. Rank and optimal computation of generic tensors. Lin. Alg. Appl., 52/53: 645-685, 1983.
42. V. Strassen. Vermeidung von Divisionen. J. Reine Angew. Math., 264: 184-202, 1973.
43. S. Winograd. On multiplication of 2 × 2 matrices, Lin. Alg. Appl., 4(4): 381-388, 1971.
44. C.N. Yang. Some exact results for the many-body problem in one dimension with repulsive delta-function interaction. Phys. Rev. Lett., 19(23): 1312-1315, 1967.
45. T. Yokonuma. Tensor Spaces and Exterior Algebra. AMS, Providence, RI, 1992.