Alan Edelman. Why are random matrix eigenvalues cool? MAA Mathfest 2002 Thursday, August 1. MIT: Dept of Mathematics, Lab for Computer Science
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1 Why are random matrix eigenvalues cool? Alan Edelman MIT: Dept of Mathematics, Lab for Computer Science MAA Mathfest 2002 Thursday, August 6/2/2004
2 Message Ingredient: Take Any important mathematics Then Randomize! This will have many applications! 6/2/2004 2
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 6/2/2004 3
4 irko s Circular Law, n=2000 Has anyone studied spacings? 6/2/2004 4
5 Wigner s Semi-Circle The classical & most famous rand eig theorem Let S = random symmetric aussian MATLAB: A=randn(n); S=(A+A )/2; Normalized eigenvalue histogram is a semi-circle Precise statements require n etc. 5
6 Wigner s Semi-Circle The classical & most famous rand eig theorem Let S = random symmetric aussian 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=; %imaginary() or real(0) for i=:s, a=randn(n)+im*sqrt(-)*randn(n);a=(a+a')/(2*sqrt(2*n*(im+))); v=eig(a)'; e=[e v]; end hold off; [m x]=hist(e,-.5:d:.5); bar(x,m*pi/(2*d*n*s)); axis('square'); axis([ ]); hold on; t=-:.0:; plot(t,sqrt(-t.^2),'r'); 6
7 Elements of Wigner s Proof Compute E(A 2k ) = mean(λ 2k ) = (2k)th moment Verify that the semicircle is the only distribution with these moments (A 2k ) = ΣA x A xy A wz A z 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! 6/2/2004 7
8 C n = 2n n + n Catalan Numbers =# ways to parenthesize (n+) objects Matrix Power Term raph (((23)4)) A 2 A 23 A 32 A 24 A 42 A 2 (((2)3)4) A 2 A 2 A 3 A 3 A 4 A 4 ((2(34))) A 2 A 23 A 34 A 43 A 32 A 2 ((2)(34)) A 2 A 2 A 3 A 34 A 43 A 3 (((23))4) A 2 A 23 A 32 A 2 A 4 A 4 = number of special paths on n departing from once Pass, (load=advance, multiply=retreat), Return to 6/2/2004 8
9 Finite Versions n=2; n=4; n=3; n=5; 9
10 How many eigenvalues of a random matrix are real? >> e=eig(randn(7)) e = i i i i >> e=eig(randn(7)) e = i i i i i i >> e=eig(randn(7)) e = i i real real 5 real 7x7 random aussian 6/2/2004 0
11 How many eigenvalues of a random matrix are real? 7 reals 5 reals 3 reals real n= /2/
12 How many eigenvalues of a random matrix are real? 7 reals 5 reals 3 reals n= These are exact real but hard to compute! New research suggests a Jack polynomial solution. 6/2/
13 How many eigenvalues of a random matrix are real? The Probability that a matrix has all real eigenvalues is exactly P n,n =2 -n(n-)/4 Proof based on Schur Form 6/2/2004 3
14 ram Schmidt (or QR) Stochastically ram Schmidt = Orthogonal Transformations to Upper Triangular Form A = Q * R (orthog * upper triangular) 6/2/2004 4
15 Orthogonal Invariance of aussians Q*randn(n,) randn(n,) If Q orthogonal Q 6/2/2004 5
16 Orthogonal Invariance Q*randn(n,) randn(n,) If Q orthogonal Q = 6/2/2004 6
17 Chi Distribution norm(randn(n,)) χ n = χ n 6/2/2004 7
18 Chi Distribution norm(randn(n,)) χ n = χ n 6/2/2004 8
19 Chi Distribution norm(randn(n,)) χ n = χ n n need not be integer 6/2/2004 9
20 6/2/
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31 6/2/2004 3
32 O χ 6 O O O O O O O O O O 6/2/
33 O χ 6 O O O O O O O O O O 6/2/
34 O χ 6 O O O O O O O O O O 6/2/
35 O χ 6 O O O O O O O O O O 6/2/
36 O χ 6 O O O O O O O O O O 6/2/
37 O χ 6 O O O O O O O O O O 6/2/
38 O χ 6 O O O O O O O O O O 6/2/
39 O χ 6 O O O O O O O O O O 6/2/
40 O χ 6 O O O O O O O O O O 6/2/
41 O χ 6 O O χ 5 O O O O O O O O O O O O 6/2/2004 4
42 O χ 6 O O χ 5 O O O O O O O O O O O O 6/2/
43 O χ 6 O O χ 5 O O O O O O O O O O O O 6/2/
44 O χ 6 O O χ 5 O O O O O O O O O O O O 6/2/
45 O χ 6 O O χ 5 O O O O O O O O O O O O 6/2/
46 O χ 6 O O χ 5 O O O O O O O O O O O O 6/2/
47 O χ 6 O O χ 5 O O O O O O O O O O O O 6/2/
48 O χ 6 O O χ 5 O O O O O O O O O O O O 6/2/
49 O χ 6 O O χ 5 O O O χ 4 O O O O O O O O O O O O 6/2/
50 O χ 6 O O χ 5 O O O χ 4 O O O O O O O O O O O O 6/2/
51 O χ 6 O O χ 5 O O O χ 4 O O O O O O O O O O O O 6/2/2004 5
52 O χ 6 O O χ 5 O O O χ 4 O O O O O O O O O O O O 6/2/
53 O χ 6 O O χ 5 O O O χ 4 O O O O O O O O O O O O 6/2/
54 O χ 6 O O χ 5 O O O χ 4 O O O O O O O O O O O O 6/2/
55 O χ 6 O O χ 5 O O O χ 4 O O O O O O O O O O O O 6/2/
56 O χ 6 O O χ 5 O O O χ 4 O O O O χ 3 O O O O O O O O O O 6/2/
57 O χ 6 O O χ 5 O O O χ 4 O O O O χ 3 O O O O O O O O O O 6/2/
58 O χ 6 O O χ 5 O O O χ 4 O O O O χ 3 O O O O O O O O O O 6/2/
59 O χ 6 O O χ 5 O O O χ 4 O O O O χ 3 O O O O O O O O O O 6/2/
60 O χ 6 O O χ 5 O O O χ 4 O O O O χ 3 O O O O O O O O O O 6/2/
61 O χ 6 O O χ 5 O O O χ 4 O O O O χ 3 O O O O O O O O O O 6/2/2004 6
62 O χ 6 O O χ 5 O O O χ 4 O O O O χ 3 O O O O O χ 2 O O O O O O 6/2/
63 O χ 6 O O χ 5 O O O χ 4 O O O O χ 3 O O O O O χ 2 O O O O O O 6/2/
64 O χ 6 O O χ 5 O O O χ 4 O O O O χ 3 O O O O O χ 2 O O O O O O 6/2/
65 O χ 6 O O χ 5 O O O χ 4 O O O O χ 3 O O O O O χ 2 O O O O O O 6/2/
66 O χ 6 O O χ 5 O O O χ 4 O O O O χ 3 O O O O O χ 2 O O O O O O 6/2/
67 O χ 6 O O χ 5 O O O χ 4 O O O O χ 3 O O O O O χ 2 O O O O O O χ 6/2/
68 O χ 6 O O χ 5 O O O χ 4 O O O O χ 3 O O O O O χ 2 O O O O O O χ 6/2/
69 Same idea: sym matrix to tridiagonal form χ 6 χ 6 χ 5 χ 5 χ 4 χ 4 χ 3 Same eigenvalue distribution as OE: O(n) storage!! χ 3 O(n) computation (potentially) χ 2 χ 2 χ χ 6/2/
70 Same idea: eneral beta χ 6β χ 6β beta: : reals 2: complexes 4: quaternions χ 5β χ 5β χ Bidiagonal 4β Version corresponds χ To Wishart 4β χmatrices of Statistics 3β χ 3β χ 2β χ 2β χ β χ β 6/2/
71 Numerical Analysis: Condition Numbers κ(a) = condition number of A If A=UΣV is the svd, then κ(a) = σ max /σ min. Alternatively, κ(a) = λ max (A A)/ λ min (A A) One number that measures digits lost in finite precision and general matrix badness Small=good Large=bad The condition of a random matrix??? 6/2/2004 7
72 Von Neumann & co. Solve Ax=b via x= (A A) - A b M A - Matrix Residual: AM-I 2 AM-I 2 < 200κ 2 n ε How should we estimate κ? Assume, as a model, that the elements of A are independent standard normals! 6/2/
73 Von Neumann & co. estimates (947-95) For a random matrix of order n the expectation value has been shown to be about n X oldstine, von Neumann we choose two different values of κ, namely n and 0n P(κ<n) 0.02 P(κ< 0n) 0.44 Bargmann, Montgomery, vn With a probability ~ κ < 0n P(κ<0n) 0.80 oldstine, von Neumann 6/2/
74 Random cond numbers, n y Distribution of κ/n = x e 2 / x 2 / x 3 x 2 Experiment with n=200 6/2/
75 Finite n n=0 n=25 n=50 n=00 6/2/
76 76 Small World Networks: & 6 degrees of separation Edelman, Eriksson, Strang Eigenvalues of A=T+PTP, P=randperm(n) Incidence matrix of graph with two superimposed cycles. = T O O O
77 77 Small World Networks: & 6 degrees of separation Edelman, Eriksson, Strang Eigenvalues of A=T+PTP, P=randperm(n) Incidence matrix of graph with two superimposed cycles. Wigner style derivation counts number of paths on a tree starting and ending at the same point (tree = no accidents!) (McKay) We first discovered the formula using the superseeker Catalan number answer d 2n- -Σ d 2j- (d-) n-j+ C n-j = T O O O
78 6/2/ The Riemann Zeta Function = = Γ = 0 ) ( ) ζ( k s u s k du e u s s 6 π ζ(2) = = K On the real line with x>, for example May be analytically extended to the complex plane, with singularity only at x=.
79 6/2/ ½ 2 3 The Riemann Hypothesis = = Γ = 0 ) ( ) ζ( k x u x k du e u x x All nontrivial roots of ζ(x) satisfy Re(x)=/2. (Trivial roots at negative even integers.)
80 ζ( x) ζ(x) along Re(x)=/2 Zeros =.5+i* The Riemann Hypothesis u x = du = Γ( x) e u k x k= ½ All nontrivial roots of ζ(x) satisfy Re(x)=/2. (Trivial roots at negative even integers.) 6/2/
81 Computation of Zeros Odlyzko s fantastic computation of 0^k+ through 0^k+0,000 for k=2,2,22. See Spacings behave like the eigenvalues of A=randn(n)+i*randn(n); S=(A+A )/2; 6/2/2004 8
82 Nearest Neighbor Spacings & Pairwise Correlation Functions 6/2/
83 Painlevé Equations 6/2/
84 Spacings Take a large collection of consecutive zeros/eigenvalues. Normalize so that average spacing =. Spacing Function = Histogram of consecutive differences (the (k+)st the kth) Pairwise Correlation Function = Histogram of all possible differences (the kth the jth) Conjecture: These functions are the same for random matrices and Riemann zeta 6/2/
85 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 6/2/
86 eneral A=LU A=UΣV T A=XΛX - Sym S=QΛQ T S=LL T S=LDL T Tridiagonal Matrix Factorization Jacobians u n-i ii (σ i2 - σ j2 ) (λ i -λ j ) 2 (λ i -λ j ) 2 n l ii n+-i d i n-i A=QR A=QS (polar) Orthogonal C S S -C Q=U V T r ii m-i (σ i +σ j ) [ ] sin(θ i + θ j )sin (θ i - θ j ) T=QΛQ T (t i+,i )/ q i 6/2/
87 Why cool? Why is numerical linear algebra cool? Mixture of theory and applications Touches many topics Easy to jump in to, but can spend a lifetime studying & researching Tons of activity in many areas Mathematics: Combinatorics, Harmonic Analysis, Integral Equations, Probability, Number Theory Applied Math: Chaotic Systems, Statistical Mechanics, Communications Theory, Radar Tracking, Nuclear Physics Applications BI HUE SUBJECT!! 6/2/
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