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随机矩阵

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概率论数学物理中,随机矩阵(英语:Random matrix)是一个矩阵值的随机变量,也就是说,一个矩阵中的所有元素都是随机变量。[1]

应用

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物理

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其他(AI、数学、统计)

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随机矩阵模型

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的矩阵,有下面的概率测度

例子,高斯模型:

参见

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阅读

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参考文献

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  1. ^ 1.0 1.1 Terence Tao 陶哲轩. Topics in random matrix theory (PDF). (原始内容 (PDF)存档于2021-05-06) (英语). 
  2. ^ Wigner, E. Characteristic vectors of bordered matrices with infinite dimensions. Annals of Mathematics. 1955, 62 (3): 548–564. JSTOR 1970079. doi:10.2307/1970079. 
  3. ^ Mehta, M.L. Random Matrices. Amsterdam: Elsevier/Academic Press. 2004. ISBN 0-12-088409-7. 
  4. ^ Bohigas, O.; Giannoni, M.J.; Schmit, Schmit. Characterization of Chaotic Quantum Spectra and Universality of Level Fluctuation Laws. Phys. Rev. Lett. 1984, 52 (1): 1–4. Bibcode:1984PhRvL..52....1B. doi:10.1103/PhysRevLett.52.1. 
  5. ^ Aaronson, Scott; Arkhipov, Alex. The computational complexity of linear optics. Theory of Computing. 2013, 9: 143–252. doi:10.4086/toc.2013.v009a004. 
  6. ^ Russell, Nicholas; Chakhmakhchyan, Levon; O'Brien, Jeremy; Laing, Anthony. Direct dialling of Haar random unitary matrices. New J. Phys. 2017, 19 (3): 033007. Bibcode:2017NJPh...19c3007R. arXiv:1506.06220可免费查阅. doi:10.1088/1367-2630/aa60ed. 
  7. ^ Random Matrix Theory and Chiral Symmetry in QCD. Annu. Rev. Nucl. Part. Sci. 2000, 50: 343–410. Bibcode:2000ARNPS..50..343V. arXiv:hep-ph/0003017可免费查阅. doi:10.1146/annurev.nucl.50.1.343. 
  8. ^ Horizon in random matrix theory, the Hawking radiation, and flow of cold atoms. Phys. Rev. Lett. October 2009, 103 (16): 166401. Bibcode:2009PhRvL.103p6401F. PMID 19905710. arXiv:0905.3533可免费查阅. doi:10.1103/PhysRevLett.103.166401. 
  9. ^ Magnetic-field asymmetry of nonlinear mesoscopic transport. Phys. Rev. Lett. September 2004, 93 (10): 106802. Bibcode:2004PhRvL..93j6802S. PMID 15447435. arXiv:cond-mat/0404387可免费查阅. doi:10.1103/PhysRevLett.93.106802. 
  10. ^ Spin torque and waviness in magnetic multilayers: a bridge between Valet-Fert theory and quantum approaches. Phys. Rev. Lett. August 2009, 103 (6): 066602. Bibcode:2009PhRvL.103f6602R. PMID 19792592. arXiv:0902.4360可免费查阅. doi:10.1103/PhysRevLett.103.066602. 
  11. ^ Callaway DJE. Random matrices, fractional statistics, and the quantum Hall effect. Phys. Rev. B. April 1991, 43 (10): 8641–8643. Bibcode:1991PhRvB..43.8641C. PMID 9996505. doi:10.1103/PhysRevB.43.8641. 
  12. ^ Correlated random band matrices: localization-delocalization transitions. Phys. Rev. E. June 2000, 61 (6 Pt A): 6278–86. Bibcode:2000PhRvE..61.6278J. PMID 11088301. arXiv:cond-mat/9911467可免费查阅. doi:10.1103/PhysRevE.61.6278. 
  13. ^ Spin-orbit coupling, antilocalization, and parallel magnetic fields in quantum dots. Phys. Rev. Lett. December 2002, 89 (27): 276803. Bibcode:2002PhRvL..89A6803Z. PMID 12513231. arXiv:cond-mat/0208436可免费查阅. doi:10.1103/PhysRevLett.89.276803. 
  14. ^ Bahcall SR. Random Matrix Model for Superconductors in a Magnetic Field. Phys. Rev. Lett. December 1996, 77 (26): 5276–5279. Bibcode:1996PhRvL..77.5276B. PMID 10062760. arXiv:cond-mat/9611136可免费查阅. doi:10.1103/PhysRevLett.77.5276. 
  15. ^ Keating, Jon. The Riemann zeta-function and quantum chaology. Proc. Internat. School of Phys. Enrico Fermi. 1993, CXIX: 145–185. ISBN 9780444815880. doi:10.1016/b978-0-444-81588-0.50008-0. 
  16. ^ Wishart, J. Generalized product moment distribution in samples. Biometrika. 1928, 20A (1–2): 32–52. doi:10.1093/biomet/20a.1-2.32. 
  17. ^ Tropp, J. User-Friendly Tail Bounds for Sums of Random Matrices. Foundations of Computational Mathematics. 2011, 12 (4): 389–434. arXiv:1004.4389可免费查阅. doi:10.1007/s10208-011-9099-z. 
  18. ^ von Neumann, J.; Goldstine, H.H. Numerical inverting of matrices of high order. Bull. Amer. Math. Soc. 1947, 53 (11): 1021–1099. doi:10.1090/S0002-9904-1947-08909-6. 
  19. ^ Edelman, A.; Rao, N.R. Random matrix theory. Acta Numerica. 2005, 14: 233–297. Bibcode:2005AcNum..14..233E. doi:10.1017/S0962492904000236. 
  20. ^ Chow, Gregory P. Analysis and Control of Dynamic Economic Systems. New York: Wiley. 1976. ISBN 0-471-15616-7. 
  21. ^ Turnovsky, Stephen. Optimal stabilization policies for stochastic linear systems: The case of correlated multiplicative and additive disturbances. Review of Economic Studies. 1976, 43 (1): 191–194. JSTOR 2296741. doi:10.2307/2296614. 
  22. ^ 22.0 22.1 Turnovsky, Stephen. The stability properties of optimal economic policies. American Economic Review. 1974, 64 (1): 136–148. JSTOR 1814888. 
  23. ^ García del Molino, Luis Carlos; Pakdaman, Khashayar; Touboul, Jonathan; Wainrib, Gilles. Synchronization in random balanced networks. Physical Review E. October 2013, 88 (4): 042824. Bibcode:2013PhRvE..88d2824G. arXiv:1306.2576可免费查阅. doi:10.1103/PhysRevE.88.042824. 
  24. ^ Rajan, Kanaka; Abbott, L. Eigenvalue Spectra of Random Matrices for Neural Networks. Physical Review Letters. November 2006, 97 (18): 188104. Bibcode:2006PhRvL..97r8104R. PMID 17155583. doi:10.1103/PhysRevLett.97.188104. 
  25. ^ Wainrib, Gilles; Touboul, Jonathan. Topological and Dynamical Complexity of Random Neural Networks. Physical Review Letters. March 2013, 110 (11): 118101. Bibcode:2013PhRvL.110k8101W. PMID 25166580. arXiv:1210.5082可免费查阅. doi:10.1103/PhysRevLett.110.118101. 
  26. ^ Muir, Dylan; Mrsic-Flogel, Thomas. Eigenspectrum bounds for semirandom matrices with modular and spatial structure for neural networks (PDF). Phys. Rev. E. 2015, 91 (4): 042808 [2020-01-13]. Bibcode:2015PhRvE..91d2808M. PMID 25974548. doi:10.1103/PhysRevE.91.042808. (原始内容 (PDF)存档于2018-07-21). 
  27. ^ Cosme Louart, Zhenyu Liao, and Romain Couillet. A RANDOM MATRIX APPROACH TO NEURAL NETWORKS (PDF). (原始内容 (PDF)存档于2020-01-13). 
  28. ^ Zhenyu Liao, Romain Couillet. The Dynamics of Learning: A Random Matrix Approach (PDF). (原始内容 (PDF)存档于2020-11-12). 
  29. ^ Jeffrey Pennington, Pratik Worah. Nonlinear random matrix theory for deep learning (PDF). (原始内容 (PDF)存档于2020-11-03). 
  30. ^ Mehta, M.L. Random Matrices. Amsterdam: Elsevier/Academic Press. 2004. ISBN 0-12-088409-7. 
  31. ^ Anderson, G.W.; Guionnet, A.; Zeitouni, O. An introduction to random matrices.. Cambridge: Cambridge University Press. 2010. ISBN 978-0-521-19452-5. 
  32. ^ Akemann, G.; Baik, J.; Di Francesco, P. The Oxford Handbook of Random Matrix Theory.. Oxford: Oxford University Press. 2011. ISBN 978-0-19-957400-1. 
  33. ^ Edelman, A.; Rao, N.R. Random matrix theory. Acta Numerica. 2005, 14: 233–297. Bibcode:2005AcNum..14..233E. doi:10.1017/S0962492904000236. 
  34. ^ Pastur, L.A. Spectra of random self-adjoint operators. Russ. Math. Surv. 1973, 28 (1): 1–67. Bibcode:1973RuMaS..28....1P. doi:10.1070/RM1973v028n01ABEH001396. 
  35. ^ Diaconis, Persi. Patterns in eigenvalues: the 70th Josiah Willard Gibbs lecture. American Mathematical Society. Bulletin. New Series. 2003, 40 (2): 155–178. MR 1962294. doi:10.1090/S0273-0979-03-00975-3. 
  36. ^ Diaconis, Persi. What is ... a random matrix?. Notices of the American Mathematical Society. 2005, 52 (11): 1348–1349 [2020-01-13]. ISSN 0002-9920. MR 2183871. (原始内容存档于2019-03-28). 
  37. ^ Wigner, E. Characteristic vectors of bordered matrices with infinite dimensions. Annals of Mathematics. 1955, 62 (3): 548–564. JSTOR 1970079. doi:10.2307/1970079. 
  38. ^ Wishart, J. Generalized product moment distribution in samples. Biometrika. 1928, 20A (1–2): 32–52. doi:10.1093/biomet/20a.1-2.32. 
  39. ^ von Neumann, J.; Goldstine, H.H. Numerical inverting of matrices of high order. Bull. Amer. Math. Soc. 1947, 53 (11): 1021–1099. doi:10.1090/S0002-9904-1947-08909-6.