Why are random matrix
eigenvalues cool?
Alan Edelman
MIT: Dept of Mathematics,
Lab for Computer Science
MAA Mathfest 2002
Thursday, August 1
6/21/2004
1
Message
™
™
™
Ingredient: Take Any important mathematics
Then Randomize!
This will have many applications!
6/21/2004
2
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
™
6/21/2004
3
Girko’s Circular Law, n=2000
Has anyone studied
spacings?
6/21/2004
4
™ The
Wigner’s Semi-Circle
classical & most famous rand eig theorem
™ Let S = random symmetric Gaussian
™ MATLAB:
™ Normalized
™ Precise
A=randn(n); S=(A+A’)/2;
eigenvalue histogram is a semi-circle
statements require n→∞ etc.
5
Wigner’s
Semi-Circle
™ The
classical & most famous rand eig theorem
™ Let S = random symmetric Gaussian
™ MATLAB:
™ Normalized
™ Precise
A=randn(n); S=(A+A’)/2;
eigenvalue histogram is a semi-circle
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
Elements of Wigner’s Proof
™ Compute
E(A2k)11= mean(λ2k) = (2k)th moment
™ Verify
that the semicircle is the only distribution
with these moments
™ (A2k)11 = ΣA1xAxy…AwzAz1 “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!
6/21/2004
7
1 ⎛ 2n ⎞
⎜⎜ ⎟⎟=#
Cn =
n +1⎝ n ⎠
Catalan Numbers
ways to “parenthesize” (n+1) objects
Matrix Power Term
Graph
(1((23)4)) A12A23A32A24A42A21
(((12)3)4) A12A21A13A31A14A41
(1(2(34))) A12A23A34A43A32A21
((12)(34)) A12A21A13A34A43A31
((1(23))4) A12A23A32A21A14A41
= number of special paths on n departing from 1 once
™ Pass 1, (load=advance, multiply=retreat), Return to 1
6/21/2004
8
n=2;
n=3;
Finite Versions
n=4;
n=5;
9
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
3 real
>> 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
1 real
>> e=eig(randn(7))
e=
-2.1633
-0.9264
-0. 3283
2.5242
1.6230 + 0.9011i
1.6230 - 0.9011i
0.5467
5 real
7x7 random Gaussian
6/21/2004
10
How many eigenvalues of a random
matrix are real?
n=7
1
7 reals
2048
355
3
−
5 reals
4096
2048
355
1087
+
3 reals −
2048
2048
4451
1085
−
1 real
4096
2048
6/21/2004
2
0.00069
2
0.08460
2
0.57727
2
0.33744
11
How many eigenvalues of a random
matrix are real?
n=7
1
2 0.00069
7 reals
2048
355
3
−
2 0.08460
5 reals
4096
2048
355
1087
+
2 0.57727
3 reals −
2048
2048
4451
1085
These are1exact
−
2 0.33744
real but hard to compute!
4096
2048
New research suggests
a Jack polynomial
solution.
6/21/2004
12
How many eigenvalues of a random
matrix are real?
™ The
Probability that a matrix has all real
eigenvalues is exactly
-n(n-1)/4
Pn,n=2
Proof based on Schur Form
6/21/2004
13
Gram Schmidt (or QR) Stochastically
•Gram Schmidt
= Orthogonal Transformations to
Upper Triangular Form
•A = Q * R (orthog * upper triangular)
6/21/2004
14
Orthogonal Invariance of Gaussians
Q*randn(n,1)
≡
randn(n,1)
If Q orthogonal
6/21/2004
Q∗
G
G
G
G
G
G
G
15
Orthogonal Invariance
Q*randn(n,1)
≡
randn(n,1)
If Q orthogonal
6/21/2004
Q∗
G
G
G
G
G
G
G
=
G
G
G
G
G
G
G
16
Chi Distribution
norm(randn(n,1))
≡
χn
6/21/2004
G
G
G
G
G
G
G
=
χn
17
Chi Distribution
norm(randn(n,1))
≡
χn
6/21/2004
G
G
G
G
G
G
G
=
χn
18
Chi Distribution
norm(randn(n,1))
≡
χn
n need not be integer
6/21/2004
G
G
G
G
G
G
G
=
χn
19
6/21/2004
G
G
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20
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G
G
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21
6/21/2004
χ7
O
O
O
O
O
O
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
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G
G
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G
G
G
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G
G
G
G
G
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G
G
22
6/21/2004
χ7
O
O
O
O
O
O
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
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G
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G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
23
6/21/2004
χ7
O
O
O
O
O
O
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
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G
G
G
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G
G
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24
6/21/2004
χ7
O
O
O
O
O
O
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
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G
G
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G
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G
G
G
G
G
G
G
G
G
G
G
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G
G
G
25
6/21/2004
χ7
O
O
O
O
O
O
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
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G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
26
6/21/2004
χ7
O
O
O
O
O
O
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
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G
G
G
G
G
G
G
G
G
G
G
G
G
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G
G
G
G
27
6/21/2004
χ7
O
O
O
O
O
O
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
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G
G
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G
G
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G
G
G
G
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G
G
G
G
28
6/21/2004
χ7
O
O
O
O
O
O
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
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G
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29
6/21/2004
χ7
O
O
O
O
O
O
G
G
G
G
G
G
G
G
G
G
G
G
G
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G
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G
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G
G
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30
6/21/2004
χ7
O
O
O
O
O
O
G
G
G
G
G
G
G
G
G
G
G
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31
6/21/2004
χ7
O
O
O
O
O
O
G
χ6
O
O
O
O
O
G
G
G
G
G
G
G
G
G
G
G
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32
6/21/2004
χ7
O
O
O
O
O
O
G
χ6
O
O
O
O
O
G
G
G
G
G
G
G
G
G
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G
G
G
G
G
G
G
G
33
6/21/2004
χ7
O
O
O
O
O
O
G
χ6
O
O
O
O
O
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
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G
G
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G
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G
G
G
G
G
G
G
G
G
34
6/21/2004
χ7
O
O
O
O
O
O
G
χ6
O
O
O
O
O
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
35
6/21/2004
χ7
O
O
O
O
O
O
G
χ6
O
O
O
O
O
G
G
G
G
G
G
G
G
G
G
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G
G
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G
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G
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G
G
G
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G
G
G
G
G
G
G
G
G
36
6/21/2004
χ7
O
O
O
O
O
O
G
χ6
O
O
O
O
O
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
37
6/21/2004
χ7
O
O
O
O
O
O
G
χ6
O
O
O
O
O
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
38
6/21/2004
χ7
O
O
O
O
O
O
G
χ6
O
O
O
O
O
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
39
6/21/2004
χ7
O
O
O
O
O
O
G
χ6
O
O
O
O
O
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
40
6/21/2004
χ7
O
O
O
O
O
O
G
χ6
O
O
O
O
O
G
G
χ5
O
O
O
O
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
41
6/21/2004
χ7
O
O
O
O
O
O
G
χ6
O
O
O
O
O
G
G
χ5
O
O
O
O
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
42
6/21/2004
χ7
O
O
O
O
O
O
G
χ6
O
O
O
O
O
G
G
χ5
O
O
O
O
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
43
6/21/2004
χ7
O
O
O
O
O
O
G
χ6
O
O
O
O
O
G
G
χ5
O
O
O
O
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
44
6/21/2004
χ7
O
O
O
O
O
O
G
χ6
O
O
O
O
O
G
G
χ5
O
O
O
O
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
45
6/21/2004
χ7
O
O
O
O
O
O
G
χ6
O
O
O
O
O
G
G
χ5
O
O
O
O
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
46
6/21/2004
χ7
O
O
O
O
O
O
G
χ6
O
O
O
O
O
G
G
χ5
O
O
O
O
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
47
6/21/2004
χ7
O
O
O
O
O
O
G
χ6
O
O
O
O
O
G
G
χ5
O
O
O
O
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
48
6/21/2004
χ7
O
O
O
O
O
O
G
χ6
O
O
O
O
O
G
G
χ5
O
O
O
O
G
G
G
χ4
O
O
O
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
49
6/21/2004
χ7
O
O
O
O
O
O
G
χ6
O
O
O
O
O
G
G
χ5
O
O
O
O
G
G
G
χ4
O
O
O
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
50
6/21/2004
χ7
O
O
O
O
O
O
G
χ6
O
O
O
O
O
G
G
χ5
O
O
O
O
G
G
G
χ4
O
O
O
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
51
6/21/2004
χ7
O
O
O
O
O
O
G
χ6
O
O
O
O
O
G
G
χ5
O
O
O
O
G
G
G
χ4
O
O
O
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
52
6/21/2004
χ7
O
O
O
O
O
O
G
χ6
O
O
O
O
O
G
G
χ5
O
O
O
O
G
G
G
χ4
O
O
O
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
53
6/21/2004
χ7
O
O
O
O
O
O
G
χ6
O
O
O
O
O
G
G
χ5
O
O
O
O
G
G
G
χ4
O
O
O
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
54
6/21/2004
χ7
O
O
O
O
O
O
G
χ6
O
O
O
O
O
G
G
χ5
O
O
O
O
G
G
G
χ4
O
O
O
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
G
55
6/21/2004
χ7
O
O
O
O
O
O
G
χ6
O
O
O
O
O
G
G
χ5
O
O
O
O
G
G
G
χ4
O
O
O
G
G
G
G
χ3
O
O
G
G
G
G
G
G
G
G
G
G
G
G
G
G
56
6/21/2004
χ7
O
O
O
O
O
O
G
χ6
O
O
O
O
O
G
G
χ5
O
O
O
O
G
G
G
χ4
O
O
O
G
G
G
G
χ3
O
O
G
G
G
G
G
G
G
G
G
G
G
G
G
G
57
6/21/2004
χ7
O
O
O
O
O
O
G
χ6
O
O
O
O
O
G
G
χ5
O
O
O
O
G
G
G
χ4
O
O
O
G
G
G
G
χ3
O
O
G
G
G
G
G
G
G
G
G
G
G
G
G
G
58
6/21/2004
χ7
O
O
O
O
O
O
G
χ6
O
O
O
O
O
G
G
χ5
O
O
O
O
G
G
G
χ4
O
O
O
G
G
G
G
χ3
O
O
G
G
G
G
G
G
G
G
G
G
G
G
G
G
59
6/21/2004
χ7
O
O
O
O
O
O
G
χ6
O
O
O
O
O
G
G
χ5
O
O
O
O
G
G
G
χ4
O
O
O
G
G
G
G
χ3
O
O
G
G
G
G
G
G
G
G
G
G
G
G
G
G
60
6/21/2004
χ7
O
O
O
O
O
O
G
χ6
O
O
O
O
O
G
G
χ5
O
O
O
O
G
G
G
χ4
O
O
O
G
G
G
G
χ3
O
O
G
G
G
G
G
G
G
G
G
G
G
G
G
G
61
6/21/2004
χ7
O
O
O
O
O
O
G
χ6
O
O
O
O
O
G
G
χ5
O
O
O
O
G
G
G
χ4
O
O
O
G
G
G
G
χ3
O
O
G
G
G
G
G
χ2
O
G
G
G
G
G
G
G
62
6/21/2004
χ7
O
O
O
O
O
O
G
χ6
O
O
O
O
O
G
G
χ5
O
O
O
O
G
G
G
χ4
O
O
O
G
G
G
G
χ3
O
O
G
G
G
G
G
χ2
O
G
G
G
G
G
G
G
63
6/21/2004
χ7
O
O
O
O
O
O
G
χ6
O
O
O
O
O
G
G
χ5
O
O
O
O
G
G
G
χ4
O
O
O
G
G
G
G
χ3
O
O
G
G
G
G
G
χ2
O
G
G
G
G
G
G
G
64
6/21/2004
χ7
O
O
O
O
O
O
G
χ6
O
O
O
O
O
G
G
χ5
O
O
O
O
G
G
G
χ4
O
O
O
G
G
G
G
χ3
O
O
G
G
G
G
G
χ2
O
G
G
G
G
G
G
G
65
6/21/2004
χ7
O
O
O
O
O
O
G
χ6
O
O
O
O
O
G
G
χ5
O
O
O
O
G
G
G
χ4
O
O
O
G
G
G
G
χ3
O
O
G
G
G
G
G
χ2
O
G
G
G
G
G
G
G
66
6/21/2004
χ7
O
O
O
O
O
O
G
χ6
O
O
O
O
O
G
G
χ5
O
O
O
O
G
G
G
χ4
O
O
O
G
G
G
G
χ3
O
O
G
G
G
G
G
χ2
O
G
G
G
G
G
G
χ1
67
6/21/2004
χ7
O
O
O
O
O
O
G
χ6
O
O
O
O
O
G
G
χ5
O
O
O
O
G
G
G
χ4
O
O
O
G
G
G
G
χ3
O
O
G
G
G
G
G
χ2
O
G
G
G
G
G
G
χ1
68
Same idea: sym matrix to tridiagonal form
G χ6
χ6 G χ5
χ5 G χ4
G
χ
χ
4
Same eigenvalue distribution3 as GOE:
O(n) storage !!
χ3 G χ2
O(n) computation (potentially)
χ2 G χ1
χ1 G
6/21/2004
69
Same idea: General beta
G χ6β beta:
1: reals 2: complexes 4: quaternions
χ6β G χ5β
χ5β GBidiagonal
χ4β Version corresponds
matrices of Statistics
G χ3β
χ4βTo Wishart
χ3β G χ2β
χ2β G χβ
χβ G
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70
™ κ(A)
Numerical Analysis:
Condition Numbers
= “condition number of A”
™ If A=UΣV’ is the svd, then κ(A) = σmax/σmin .
™ 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???
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71
Von Neumann & co.
™
Solve Ax=b via x= (A’A) -1A’ b
M ≈A-1
™
Matrix Residual: ||AM-I||2
™
||AM-I||2< 200κ2 n ε
≈
™
How should we estimate κ?
™
Assume, as a model, that the elements of A are
independent standard normals!
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72
Von Neumann & co. estimates
(1947-1951)
™
“For a ‘random matrix’ of order n the expectation value
has been shown to be about X
n” ∞
Goldstine, von Neumann
™
“… we choose two different values of κ, namely n and
√10n” P(κ<n)
≈0.02
Bargmann, Montgomery, vN
P(κ< √10n)≈0.44
™
“With a probability ~1 … κ < 10n”
P(κ<10n) ≈0.80
6/21/2004
Goldstine, von Neumann
73
Random cond numbers, n→∞
Distribution of κ/n
2 x + 4 −2 / x −2 / x 2
y=
e
3
x
6/21/2004
Experiment with n=200
74
Finite n
n=10
n=25
n=50
n=100
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75
Small World Networks:
& 6 degrees of separation
Edelman, Eriksson, Strang
™ Eigenvalues of A=T+PTP’, P=randperm(n)
1⎤
⎡0 1
Incidence matrix of graph with two
⎥
⎢1 0 1
⎥
⎢
⎥
⎢
1 0 1
⎥
⎢
superimposed cycles.
1 0 O
⎥
T=⎢
™
⎢
⎢
⎢
⎢
⎢1
⎣
O O 1
1
⎥
0 1 ⎥
⎥
1 0 1⎥
1 0⎥⎦
76
Small World Networks:
& 6 degrees of separation
Edelman, Eriksson, Strang
™ Eigenvalues of A=T+PTP’, P=randperm(n)
1⎤
⎡0 1
Incidence matrix of graph with two
⎥
⎢1 0 1
⎥
⎢
⎥
⎢
1 0 1
⎥
⎢
superimposed cycles.
1 0 O
⎥
T=⎢
™
⎢
⎢
⎢
⎢
⎢1
⎣
O O 1
1
⎥
0 1 ⎥
⎥
1 0 1⎥
1 0⎥⎦
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
77
™ Catalan number answer d2n-1-Σ d2j-1(d-1)n-j+1Cn-j
™
The Riemann Zeta Function
1
ζ( s ) =
Γ (s)
∞
∫
0
s −1
u
du =
u
e −1
∞
∑
k =1
1
ks
On the real line with x>1, for example
2
1 1 1
π
ζ(2) = 1 + 2 + 2 + 2 + K =
2 3 4
6
May be analytically extended to the complex plane,
with singularity only at x=1.
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78
The Riemann Hypothesis
∞
x −1
-1
0
∞
u
1
1
du
ζ( x ) =
=
∑
u
x
∫
Γ( x ) 0 e − 1
k =1 k
-3
-2
½ 1
2
3
All nontrivial roots of ζ(x) satisfy Re(x)=1/2.
(Trivial roots at negative even integers.)
6/21/2004
79
=.5+i*
|ζ(x)| along Re(x)=1/2 Zeros
14.134725142
21.022039639
The Riemann Hypothesis
25.010857580
∞
x −1
∞
30.424876126
u
1
1
32.935061588
du
ζ( x ) =
=
∑
u
x
∫
37.586178159
Γ( x ) 0 e − 1 40.918719012
k =1 k
-3
-2
-1
0
43.327073281
48.005150881
49.773832478
½ 152.970321478
2
3
56.446247697
59.347044003
All nontrivial roots of ζ(x) satisfy Re(x)=1/2.
(Trivial roots at negative even integers.)
6/21/2004
80
Computation of Zeros
™ Odlyzko’s
fantastic computation of 10^k+1
through 10^k+10,000 for k=12,21,22.
See http://www.research.att.com/~amo/zeta_tables/
Spacings behave like the eigenvalues of
A=randn(n)+i*randn(n); S=(A+A’)/2;
6/21/2004
81
Nearest Neighbor Spacings &
Pairwise Correlation Functions
6/21/2004
82
Painlevé Equations
6/21/2004
83
Spacings
Take a large collection of consecutive zeros/eigenvalues.
™ Normalize so that average spacing = 1.
™ Spacing Function = Histogram of consecutive differences
(the (k+1)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
™
6/21/2004
84
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
™
6/21/2004
85
Matrix Factorization Jacobians
General
∏ riim-i
∏ uiin-i
A=LU
A=QR
A=UΣVT ∏ (σi2- σj2) A=QS (polar) ∏ (σi+σj)
A=XΛX-1 ∏ (λi-λj)2
Sym
S=QΛQT
S=LLT
S=LDLT
Orthogonal
∏ (λi-λj)
2n∏ liin+1-i Q=U C S VT ∏sin(θ + θ )sin (θ - θ )
i
j
i
j
S -C
∏ din-i
[ ]
Tridiagonal
T=QΛQT ∏ (ti+1,i)/ ∏qi
6/21/2004
86
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
™ BIG HUGE SUBJECT!!
™
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87