maamathfest2002

maamathfest2002 - Why are random matrix eigenvalues cool?...

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6/21/2004 1 Why are random matrix eigenvalues cool? Alan Edelman MIT: Dept of Mathematics, Lab for Computer Science MAA Mathfest 2002 Thursday, August 1
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6/21/2004 2 Message ± Ingredient: Take Any important mathematics ± Then Randomize! ± This will have many applications!
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6/21/2004 3 Some fun tidbits ± The circular law ± The semi-circular law ± Infinite vs finite ± How many are real? ± Stochastic Numerical Algorithms ± Condition Numbers ± Small networks ± Riemann Zeta Function ± Matrix Jacobians
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6/21/2004 4 Girko’s Circular Law, n=2000 Has anyone studied spacings?
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5 Wigner’s Semi-Circle ± ± Let S = random symmetric Gaussian ± MATLAB: A=randn(n); S=(A+A’)/2; ± Normalized eigenvalue histogram is a semi-circle ± Precise statements require n →∞ etc.
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6 Wigner’s Semi-Circle ± ± Let S = random symmetric Gaussian ± MATLAB: A=randn(n); S=(A+A’)/2; ± Normalized eigenvalue histogram is a semi-circle ± Precise statements require n →∞ etc. n=20; s=30000; d=.05; %matrix size, samples, sample dist e=[]; %gather up eigenvalues im=1; %imaginary(1) or real(0) for i=1:s, a=randn(n)+im*sqrt(-1)*randn(n);a=(a+a')/(2*sqrt(2*n*(im+1))); v=eig(a)'; e=[e v]; end hold off; [m x]=hist(e,-1.5:d:1.5); bar(x,m*pi/(2*d*n*s)); axis('square'); axis([-1.5 1.5 -1 2]); hold on; t=-1:.01:1; plot(t,sqrt(1-t.^2),'r');
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6/21/2004 7 Elements of Wigner’s Proof ± Compute E(A 2k ) 11 = mean( λ 2k ) = (2k)th moment ± Verify that the semicircle is the only distribution with these moments ± (A 2k ) 11 = Σ A 1x A xy …A wz A z1 “paths” of length 2k ± Need only count number of special paths of length 2k on k objects (all other terms 0 or negligible!) ± This is a Catalan Number!
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6/21/2004 8 Catalan Numbers =# ways to “parenthesize” (n+1) objects Matrix Power Term Graph (1((23)4)) A 12 A 23 A 32 A 24 A 42 A 21 (((12)3)4) A 12 A 21 A 13 A 31 A 14 A 41 (1(2(34))) A 12 A 23 A 34 A 43 A 32 A 21 ((12)(34)) A 12 A 21 A 13 A 34 A 43 A 31 ((1(23))4) A 12 A 23 A 32 A 21 A 14 A 41 = number of special paths on n departing from 1 once ± Pass 1, (load=advance, multiply=retreat), Return to 1 + = n n n n 2 1 1 C
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9 Finite Versions n=2; n=4; n=3; n=5;
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6/21/2004 10 How many eigenvalues of a random matrix are real? >> e=eig(randn(7)) e = 1.9771 1.3442 0.6316 -1.1664 + 1.3504i -1.1664 - 1.3504i -2.1461 + 0.7288i -2.1461 - 0.7288i >> e=eig(randn(7)) e = -2.0767 + 1.1992i -2.0767 - 1.1992i 2.9437 0.0234 + 0.4845i 0.0234 - 0.4845i 1.1914 + 0.3629i 1.1914 - 0.3629i >> e=eig(randn(7)) e = -2.1633 -0.9264 -0. 3283 2.5242 1.6230 + 0.9011i 1.6230 - 0.9011i 0.5467 7x7 random Gaussian 3 real 1 real 5 real
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11 How many eigenvalues of a random matrix are real? 2
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This note was uploaded on 11/07/2011 for the course AERO 16.26 taught by Professor Dimitribertsekas during the Fall '02 term at MIT.

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maamathfest2002 - Why are random matrix eigenvalues cool?...

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