1. Advances in Random Matrix Theory: Let there be tools
Alan Edelman
Brian Sutton, Plamen Koev,
Ioana Dumitriu,
Raj Rao and others
MIT: Dept of Mathematics,
Computer Science AI Laboratories
World Congress, Bernoulli Society
Barcelona, Spain
Wednesday July 28, 2004
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2. Tools So many applications …
Random matrix theory: catalyst for 21st century special functions, analytical techniques, statistical techniques
In addition to mathematics and papers
Need tools for the novice!
Need tools for the engineers!
Need tools for the specialists!
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3. Themes of this talk Tools for general beta
What is beta? Think of it as a measure of (inverse) volatility in “classical” random matrices.
Tools for complicated derived random matrices
Tools for numerical computation and simulation
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4. Wigner’s Semi-Circle The classical & most famous rand eig theorem
Let S = random symmetric Gaussian
MATLAB: A=randn(n); S=(A+A’)/2;
S known as the Gaussian Orthogonal Ensemble
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([ ]); hold on;
t=-1:.01:1; plot(t,sqrt(1-t.^2),'r');

5. G 6 5 4 3 2 1 sym matrix to tridiagonal form
Same eigenvalue distribution as GOE:
O(n) storage !! O(n2) compute
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6. G 6 5 4 3 2  General beta beta:
1: reals 2: complexes 4: quaternions
Bidiagonal Version corresponds
To Wishart matrices of Statistics
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7. Tools Motivation: A condition number problem
Jack & Hypergeometric of Matrix Argument
MOPS: Ioana Dumitriu’s talk
The Polynomial Method
The tridiagonal numerical 109 trick
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8. Tools Motivation: A condition number problem
Jack & Hypergeometric of Matrix Argument
MOPS: Ioana Dumitriu’s talk
The Polynomial Method
The tridiagonal numerical 109 trick
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9. Numerical Analysis: Condition Numbers
(A) = “condition number of A”
If A=UV’ is the svd, then (A) = max/min .
Alternatively, (A) =  max (A’A)/ 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???
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10. Von Neumann & co. estimates (1947-1951)
“For a ‘random matrix’ of order n the expectation value has been shown to be about n”
Goldstine, von Neumann
“… we choose two different values of , namely n and 10n”
Bargmann, Montgomery, vN
“With a probability ~1 …  < 10n”
X 
P(<n) 0.02
P(< 10n)0.44
P(<10n) 0.80
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11. Random cond numbers, n
Distribution of /n
Experiment with n=200
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12. Finite n
n= n=25
n= n=100
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13. Condition Number Distributions
P(κ/n > x) ≈ 2/x
P(κ/n2 > x) ≈ 4/x2
Real n x n, n∞
Generalizations:
β: 1=real, 2=complex
finite matrices
rectangular: m x n
Complex n x n, n∞
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14. Condition Number Distributions
P(κ/n > x) ≈ 2/x
P(κ/n2 > x) ≈ 4/x2
Square, n∞: P(κ/nβ > x) ≈ (2ββ-1/Γ(β))/xβ (All Betas!!)
General Formula: P(κ>x) ~Cµ/x β(n-m+1),
where µ= β(n-m+1)/2th moment of the largest eigenvalue of
Wm-1,n+1 (β)
and C is a known geometrical constant.
Density for the largest eig of W is known in terms of
1F1((β/2)(n+1), ((β/2)(n+m-1); -(x/2)Im-1) from which µ is available
Tracy-Widom law applies probably all beta for large m,n.
Johnstone shows at least beta=1,2.
Real n x n, n∞
Complex n x n, n∞
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15. Tools Motivation: A condition number problem
Jack & Hypergeometric of Matrix Argument
MOPS: Ioana Dumitriu’s talk
The Polynomial Method
The tridiagonal numerical 109 trick
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16. ∫ pκ(x)pλ(x) Δ(x)β ∏i w(xi)dxi = δκλ
Multivariate Orthogonal Polynomials & Hypergeometrics of Matrix Argument
Ioana Dumitriu’s talk
The important special functions of the 21st century
Begin with w(x) on I
∫ pκ(x)pλ(x) Δ(x)β ∏i w(xi)dxi = δκλ
Jack Polynomials orthogonal for w=1 on the unit circle. Analogs of xm
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17. Multivariate Hypergeometric Functions
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18. Multivariate Hypergeometric Functions
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19. Wishart (Laguerre) Matrices!
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20. Plamen’s clever idea
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21. Tools Motivation: A condition number problem
Jack & Hypergeometric of Matrix Argument
MOPS: Ioana Dumitriu’s talk
The Polynomial Method
The tridiagonal numerical 109 trick
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22. Mops (Dumitriu etc.) Symbolic
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23. Symbolic MOPS applications
A=randn(n); S=(A+A’)/2; trace(S^4)
det(S^3)
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24. Symbolic MOPS applications
β=3; hist(eig(S))
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25. Smallest eigenvalue statistics
A=randn(m,n); hist(min(svd(A).^2))
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26. Tools Motivation: A condition number problem
Jack & Hypergeometric of Matrix Argument
MOPS: Ioana Dumitriu’s talk
The Polynomial Method -- Raj!
The tridiagonal numerical 109 trick
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27. Courtesy of the Polynomial Method
RM Tool – Raj!
Courtesy of the Polynomial Method
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28. 1/18/2019

29. Tools Motivation: A condition number problem
Jack & Hypergeometric of Matrix Argument
MOPS: Ioana Dumitriu’s talk
The Polynomial Method
The tridiagonal numerical 109 trick
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30. Everyone’s Favorite Tridiagonal-2 1 1 n2 … … … … … d2
dx2
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31. Everyone’s Favorite Tridiagonal-2 1 G 1
(βn)1/2 1 n2 … … … + … … dW
β1/2 d2
dx2 +
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32. Stochastic Operator Limit, dW β 2 x dx d + - ,
N(0,2) χ n β 2 1 ~ H 2) (n 1) ÷ ø ö ç è æ - … … … , G β 2 H n +
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33. Largest Eigenvalue Plots
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34. MATLAB beta=1; n=1e9; opts.disp=0;opts.issym=1;
alpha=10; k=round(alpha*n^(1/3)); % cutoff parameters
d=sqrt(chi2rnd( beta*(n:-1:(n-k-1))))';
H=spdiags( d,1,k,k)+spdiags(randn(k,1),0,k,k);
H=(H+H')/sqrt(4*n*beta);
eigs(H,1,1,opts)
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35. Tricks to get O(n9) speedup
Sparse matrix storage (Only O(n) storage is used)
Tridiagonal Ensemble Formulas (Any beta is available due to the tridiagonal ensemble)
The Lanczos Algorithm for Eigenvalue Computation ( This allows the computation of the extreme eigenvalue faster than typical general purpose eigensolvers.)
The shift-and-invert accelerator to Lanczos and Arnoldi (Since we know the eigenvalues are near 1, we can accelerate the convergence of the largest eigenvalue)
The ARPACK software package as made available seamlessly in MATLAB (The Arnoldi package contains state of the art data structures and numerical choices.)
The observation that if k = 10n1/3 , then the largest eigenvalue is determined numerically by the top k × k segment of n. (This is an interesting mathematical statement related to the decay of the Airy function.)
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36. Random matrix tools!
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