{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

maamathfest2002

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

This preview shows pages 1–12. Sign up to view the full content.

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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
6/21/2004 2 Message ± Ingredient: Take Any important mathematics ± Then Randomize! ± This will have many applications!
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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
6/21/2004 4 Girko’s Circular Law, n=2000 Has anyone studied spacings?
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.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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');
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!

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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
9 Finite Versions n=2; n=4; n=3; n=5;

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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
11 How many eigenvalues of a random matrix are real? 2

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 87

maamathfest2002 - Why are random matrix eigenvalues cool...

This preview shows document pages 1 - 12. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online