hermite , laguerre, jacobi listen to random matrix theory it’s trying to tell us something
DESCRIPTION
Hermite , Laguerre, Jacobi Listen to Random Matrix Theory It’s trying to tell us something. Alan Edelman Mathematics February 24, 2014. Hermite , Laguerre , and Jacobi. Hermite 1822-1901. Laguerre 1834-1886. Jacobi 1804-1851. - PowerPoint PPT PresentationTRANSCRIPT
![Page 1: Hermite , Laguerre, Jacobi Listen to Random Matrix Theory It’s trying to tell us something](https://reader035.vdocuments.mx/reader035/viewer/2022062314/56813432550346895d9b2162/html5/thumbnails/1.jpg)
Hermite, Laguerre, JacobiListen to Random Matrix TheoryIt’s trying to tell us something
Alan EdelmanMathematics
February 24, 2014
![Page 2: Hermite , Laguerre, Jacobi Listen to Random Matrix Theory It’s trying to tell us something](https://reader035.vdocuments.mx/reader035/viewer/2022062314/56813432550346895d9b2162/html5/thumbnails/2.jpg)
2/49
Hermite, Laguerre, and Jacobi
Hermite1822-1901
Laguerre1834-1886
Jacobi1804-1851
![Page 3: Hermite , Laguerre, Jacobi Listen to Random Matrix Theory It’s trying to tell us something](https://reader035.vdocuments.mx/reader035/viewer/2022062314/56813432550346895d9b2162/html5/thumbnails/3.jpg)
3/49
An Intriguing Mathematical Tour
Sometimes out of my comfort zone
Opportunities Abound
![Page 4: Hermite , Laguerre, Jacobi Listen to Random Matrix Theory It’s trying to tell us something](https://reader035.vdocuments.mx/reader035/viewer/2022062314/56813432550346895d9b2162/html5/thumbnails/4.jpg)
4/49
MATH MATLAB Probability Density Remark
Standard Normal randn()
Chi-Squared chi2rnd(v)norm(randn(v,1))^
2
Beta Distribution
betarnd(a,b)
x=chi2rnd(2a)y=chi2rnd(2b)
x/(x+y)
Scalar Random Variables (n=1)
![Page 5: Hermite , Laguerre, Jacobi Listen to Random Matrix Theory It’s trying to tell us something](https://reader035.vdocuments.mx/reader035/viewer/2022062314/56813432550346895d9b2162/html5/thumbnails/5.jpg)
5/49
MATH MATLAB Probability Density Remark
Standard Normal randn()
Chi-Squared chi2rnd(v)norm(randn(v,1))^
2
Beta Distribution
betarnd(a,b)
x=chi2rnd(2a)y=chi2rnd(2b)
x/(x+y)
Scalar Random Variables (n=1)
Note for integer v
![Page 6: Hermite , Laguerre, Jacobi Listen to Random Matrix Theory It’s trying to tell us something](https://reader035.vdocuments.mx/reader035/viewer/2022062314/56813432550346895d9b2162/html5/thumbnails/6.jpg)
6/49
MATH MATLAB Probability Density Remark
Standard Normal randn()
Chi-Squared chi2rnd(v)norm(randn(v,1))^
2
Beta Distribution
betarnd(a,b)
x=chi2rnd(2a)y=chi2rnd(2b)
x/(x+y)
Scalar Random Variables (n=1)
![Page 7: Hermite , Laguerre, Jacobi Listen to Random Matrix Theory It’s trying to tell us something](https://reader035.vdocuments.mx/reader035/viewer/2022062314/56813432550346895d9b2162/html5/thumbnails/7.jpg)
7/49
MATH MATLAB Probability Density Remark
Standard Normal randn()
Chi-Squared chi2rnd(v)norm(randn(v,1))^
2
Beta Distribution
betarnd(a,b)
x=chi2rnd(2a)y=chi2rnd(2b)
x/(x+y)
Hermite
Laguerre
Jacobi
Scalar Random Variables (n=1)
![Page 8: Hermite , Laguerre, Jacobi Listen to Random Matrix Theory It’s trying to tell us something](https://reader035.vdocuments.mx/reader035/viewer/2022062314/56813432550346895d9b2162/html5/thumbnails/8.jpg)
8/49
Random Matrices
Ensembles RMs MATLAB Joint Eigenvalue Density
HermiteGaussian
EnsemblesWigner (1955)
G=randn(n,n)S=(G+G’)/2
Symmetric
LaguerreWishart Matrices
(1928)G=randn(m,n)
W=(G’*G)/nPositive Definite
JacobiMANOVAMatrices
(1939)
W1=Wishart(m1,n)
W2=Wishart(m2,n)
J=W1/(W1+W2)
(Morally)Symmetric
0 < J < I
![Page 9: Hermite , Laguerre, Jacobi Listen to Random Matrix Theory It’s trying to tell us something](https://reader035.vdocuments.mx/reader035/viewer/2022062314/56813432550346895d9b2162/html5/thumbnails/9.jpg)
9/49
Three biggies in numerical linear algebra: eig, svd, gsvd
Random Matrix Algorithm MATLAB
Hermite GaussianEnsembles eig
G=randn(n,n)S=(G+G’)/2
eig(S)
Laguerre Wishart svd svd(randn(m,n))
Jacobi MANOVA gsvd gsvd(randn(m1,n),randn(m2,n))
![Page 10: Hermite , Laguerre, Jacobi Listen to Random Matrix Theory It’s trying to tell us something](https://reader035.vdocuments.mx/reader035/viewer/2022062314/56813432550346895d9b2162/html5/thumbnails/10.jpg)
10/49
The Jacobi Ensemble:Geometric Interpretation
• Take reference n≤m dimensional subspace of Rm
• Take RANDOM n≤m dimensional subspace of Rm
• The shadow of the unit ball in the random subspace when projected onto the reference subspace is an ellipsoid
• The semi-axes lengths are the Jacobi ensemble cosines. (MANOVA Convention=Squared cosines)
![Page 11: Hermite , Laguerre, Jacobi Listen to Random Matrix Theory It’s trying to tell us something](https://reader035.vdocuments.mx/reader035/viewer/2022062314/56813432550346895d9b2162/html5/thumbnails/11.jpg)
11/49
X
Y
n-dim subspace =span( )
m=m1+m2 dimensionsn ≤ m
(m1 dim)
(m2 dim)
X
Y
c 1u 1
s1v1
π π
c1u1
s 1v1
Flattened View Expanded Viewsubspaces represented by
lines planes
c1u1s1v1
( )
c1u1s1v1
( )
GSVD(A,B)
Ex 1: Random line in R2 through 0:On the x axis: c
On the z axis: s
A,B have n columns
AB
Ex 2: Random plane in R4 through 0:On xy plane: c1,c2
On zw plane: s1,s2
![Page 12: Hermite , Laguerre, Jacobi Listen to Random Matrix Theory It’s trying to tell us something](https://reader035.vdocuments.mx/reader035/viewer/2022062314/56813432550346895d9b2162/html5/thumbnails/12.jpg)
12/49
X
Y
n-dim subspace =span( )
m=m1+m2 dimensionsn ≤ m
(m1 dim)
(m2 dim)
X
Y
c 1u 1
s1v1
π π
c1u1
s 1v1
Flattened View Expanded Viewsubspaces represented by
lines planes
c1u1s1v1
( )
c1u1s1v1
( )
GSVD(A,B)
Ex 4: Random plane in R3 through 0:On the xy plane: c and 1 (one axis in the xy plane) On the z axis: s
A,B have n columns
AB
Ex 3: Random line in R3 through 0:On the xy plane: c and 0
On the z axis: s
![Page 13: Hermite , Laguerre, Jacobi Listen to Random Matrix Theory It’s trying to tell us something](https://reader035.vdocuments.mx/reader035/viewer/2022062314/56813432550346895d9b2162/html5/thumbnails/13.jpg)
13/49
Infinite Random Matrix Theory& Gil Strang’s favorite matrix
![Page 14: Hermite , Laguerre, Jacobi Listen to Random Matrix Theory It’s trying to tell us something](https://reader035.vdocuments.mx/reader035/viewer/2022062314/56813432550346895d9b2162/html5/thumbnails/14.jpg)
14/49
Limit Laws for Eigenvalue Histograms
Law Formula
Hermite Semicircle LawWigner 1955
Free CLT
Laguerre Marcenko-Pastur Law
1967
Jacobi Wachter Law 1980
Too Small
![Page 15: Hermite , Laguerre, Jacobi Listen to Random Matrix Theory It’s trying to tell us something](https://reader035.vdocuments.mx/reader035/viewer/2022062314/56813432550346895d9b2162/html5/thumbnails/15.jpg)
15/49
Three big laws: Toeplitz+boundary
That’s pretty special!Corresponds to 2nd order differences with boundary
Anshelevich, Młotkowski(2010) (Free Meixner)E, Dubbs (2014)
Law Equilibrium Measure
Hermite Semicircle Law 1955
Free CLT
x=ay=b
Laguerre Marcenko-Pastur Law
1967
x=parametery=b
Jacobi Wachter Law 1980
x=parametery=parameter
Free Poisson
Free Binomial
![Page 16: Hermite , Laguerre, Jacobi Listen to Random Matrix Theory It’s trying to tell us something](https://reader035.vdocuments.mx/reader035/viewer/2022062314/56813432550346895d9b2162/html5/thumbnails/16.jpg)
16/49
Example Chebfun Lanczos RunVerbatim from Pedro Gonnet’s November 2011 Run
Lanczos
Thanks toBernie Wang
![Page 17: Hermite , Laguerre, Jacobi Listen to Random Matrix Theory It’s trying to tell us something](https://reader035.vdocuments.mx/reader035/viewer/2022062314/56813432550346895d9b2162/html5/thumbnails/17.jpg)
17/49
Why are Hermite, Laguerre, Jacobi
all over mathematics?
I’m still wondering.
![Page 18: Hermite , Laguerre, Jacobi Listen to Random Matrix Theory It’s trying to tell us something](https://reader035.vdocuments.mx/reader035/viewer/2022062314/56813432550346895d9b2162/html5/thumbnails/18.jpg)
18/49
Varying Inconsistent Definitions ofClassical Orthogonal Polynomials
• Hermite, Laguerre, and Jacobi Polynomials • “There is no generally accepted definition of classical
orthogonal polynomials, but …” (Walter Gautschi)• Orthogonal polynomials whose derivatives are also orthogonal
polynomials (Wikipedia: Honine, Hahn)• Hermite, Laguerre, Bessel, and Jacobi … are called collectively
the “classical orthogonal polynomials (L. Miranian)• Orthogonal Polynomials that are eigenfunctions of a fixed 2nd-
order linear differential operator (Bochner, Grünbaum,Haine)• All polynomials in the Askey scheme
![Page 19: Hermite , Laguerre, Jacobi Listen to Random Matrix Theory It’s trying to tell us something](https://reader035.vdocuments.mx/reader035/viewer/2022062314/56813432550346895d9b2162/html5/thumbnails/19.jpg)
19/49
chart from Temme, et. al
Askey Scheme
![Page 20: Hermite , Laguerre, Jacobi Listen to Random Matrix Theory It’s trying to tell us something](https://reader035.vdocuments.mx/reader035/viewer/2022062314/56813432550346895d9b2162/html5/thumbnails/20.jpg)
20/49
• The differential operator has the form
where Q(x) is (at most) quadratic and L(x) is linear
• Possess a Rodrigues formula (W(x)=weight)
• Pearson equation for the weight function itself:
Varying Inconsistent Definitions ofClassical Orthogonal Polynomials
![Page 21: Hermite , Laguerre, Jacobi Listen to Random Matrix Theory It’s trying to tell us something](https://reader035.vdocuments.mx/reader035/viewer/2022062314/56813432550346895d9b2162/html5/thumbnails/21.jpg)
21/49
Yet more properties
• Sheffer sequence ( for linear operator Q)• Hermite, Laguerre, and Jacobi are Sheffer• not sure what other orthogonal polynomials are Sheffer
• Appell Sequence must be Sheffer
• Hermite (not any other orthogonal polynomial)
![Page 22: Hermite , Laguerre, Jacobi Listen to Random Matrix Theory It’s trying to tell us something](https://reader035.vdocuments.mx/reader035/viewer/2022062314/56813432550346895d9b2162/html5/thumbnails/22.jpg)
22/49
All the definitions are formulaic• Formulas are concrete, useful, and departure points
for many properties, but they don’t feel like they explain a mathematical core
Where else might we look?
Anything more structural?
![Page 23: Hermite , Laguerre, Jacobi Listen to Random Matrix Theory It’s trying to tell us something](https://reader035.vdocuments.mx/reader035/viewer/2022062314/56813432550346895d9b2162/html5/thumbnails/23.jpg)
23/49
A View towards Structure
• Hermite: Symmetric Eigenproblem: (Sym Matrices)/(Orthogonal matrices)
(Eigenvalues )
• Laguerre: SVD (Orthogonals) \ (m x n matrices) / (Orthogonals)
(Singular Values )
• Jacobi: GSVD(Grassmann Manifold)/(Stiefel m1 x Stiefel m2)
(Cosine/Sine pairs )
KAK Decompositions?
![Page 24: Hermite , Laguerre, Jacobi Listen to Random Matrix Theory It’s trying to tell us something](https://reader035.vdocuments.mx/reader035/viewer/2022062314/56813432550346895d9b2162/html5/thumbnails/24.jpg)
24/49
Homogeneous Spaces
• Take a Lie Group and quotient out a subgroup
• The subgroup is itself an open subgroup of the fixed points of an involution
Symmetric Space
![Page 25: Hermite , Laguerre, Jacobi Listen to Random Matrix Theory It’s trying to tell us something](https://reader035.vdocuments.mx/reader035/viewer/2022062314/56813432550346895d9b2162/html5/thumbnails/25.jpg)
25/49
Symmetric Space Charts
![Page 26: Hermite , Laguerre, Jacobi Listen to Random Matrix Theory It’s trying to tell us something](https://reader035.vdocuments.mx/reader035/viewer/2022062314/56813432550346895d9b2162/html5/thumbnails/26.jpg)
26/49
Circular Ensembles
Jacobi: m1=n
Jacobi: β=4, m1=½,1½, m2= ½Jacobi: β=1,m1=n+1,m2=n+1
HaaralsoJacobi
HermiteWhat are these?
Laguerre I’m told?
![Page 27: Hermite , Laguerre, Jacobi Listen to Random Matrix Theory It’s trying to tell us something](https://reader035.vdocuments.mx/reader035/viewer/2022062314/56813432550346895d9b2162/html5/thumbnails/27.jpg)
27/49
Random Matrix Story Clearly Lined up with Symmetric Spaces (Hermite, Circular)
Symmetric Spaces fall a little short (where are the rest of the Jacobi’s???)
also a Jacobi
What are these?Some must be Laguerre
(Zirnbauer, Dueñez)
![Page 28: Hermite , Laguerre, Jacobi Listen to Random Matrix Theory It’s trying to tell us something](https://reader035.vdocuments.mx/reader035/viewer/2022062314/56813432550346895d9b2162/html5/thumbnails/28.jpg)
28/49
Coxeter Groups?
• Symmetry group of regular polyhedra• Weyl groups of simple Lie Algebras
Foundation for structure
![Page 29: Hermite , Laguerre, Jacobi Listen to Random Matrix Theory It’s trying to tell us something](https://reader035.vdocuments.mx/reader035/viewer/2022062314/56813432550346895d9b2162/html5/thumbnails/29.jpg)
29/49
Macdonald’s Integral form of Selberg’s Integral
• Integrals can arise in random matrix theory• AnHermite Bn &DnTwo special cases of Laguerre• Connection to RMT, Very Structural, but does
not line up
![Page 30: Hermite , Laguerre, Jacobi Listen to Random Matrix Theory It’s trying to tell us something](https://reader035.vdocuments.mx/reader035/viewer/2022062314/56813432550346895d9b2162/html5/thumbnails/30.jpg)
30/49
Graph Theory
• Hermite: • Random Complete Graph• Incidence Matrix is the semicircle law
• Laguerre: • Random Bipartite Graph• Incidence Matrix is Marcenko-Pastur law
• Jacobi: • Random d-regular graph (McKay)• Incidence Matrix is a special case of a Wachter law
![Page 31: Hermite , Laguerre, Jacobi Listen to Random Matrix Theory It’s trying to tell us something](https://reader035.vdocuments.mx/reader035/viewer/2022062314/56813432550346895d9b2162/html5/thumbnails/31.jpg)
31/49
Quantum MechanicsAnalytically Solvable?
• Hermite: Harmonic Oscillator• Laguerre: Radial Part of Hydrogen
• Morse Oscillator
• Jacobi: Angular part of Hydrogen is Legendre• Hyperbolic Rosen-Morse Potential
Thanks to Jiahao Chen regarding Derizinsky,Wrochna
![Page 32: Hermite , Laguerre, Jacobi Listen to Random Matrix Theory It’s trying to tell us something](https://reader035.vdocuments.mx/reader035/viewer/2022062314/56813432550346895d9b2162/html5/thumbnails/32.jpg)
32/49
Representations of Lie Algebras• Hermite Heisenberg group H_3• Laguerre Third Order Triangular Matrices• Jacobi Unimodular quasi-unitary group
In the orthogonal polynomial basis, the tridiagonal matrix and its pieces can be represented as simple differential operators
![Page 33: Hermite , Laguerre, Jacobi Listen to Random Matrix Theory It’s trying to tell us something](https://reader035.vdocuments.mx/reader035/viewer/2022062314/56813432550346895d9b2162/html5/thumbnails/33.jpg)
33/49
Wigner and Narayana
• Marcenko-Pastur = Limiting Density for Laguerre• Moments are Narayana Polynomials!• Narayana probably would not have known
[Wigner, 1957]
NarayanaPhoto
Unavailable
(Narayana was 27)
![Page 34: Hermite , Laguerre, Jacobi Listen to Random Matrix Theory It’s trying to tell us something](https://reader035.vdocuments.mx/reader035/viewer/2022062314/56813432550346895d9b2162/html5/thumbnails/34.jpg)
34/49
Cool PyramidNarayana everywhere!
![Page 35: Hermite , Laguerre, Jacobi Listen to Random Matrix Theory It’s trying to tell us something](https://reader035.vdocuments.mx/reader035/viewer/2022062314/56813432550346895d9b2162/html5/thumbnails/35.jpg)
35/49
Cool Pyramid
![Page 36: Hermite , Laguerre, Jacobi Listen to Random Matrix Theory It’s trying to tell us something](https://reader035.vdocuments.mx/reader035/viewer/2022062314/56813432550346895d9b2162/html5/thumbnails/36.jpg)
36/49
Cool Pyramid
![Page 37: Hermite , Laguerre, Jacobi Listen to Random Matrix Theory It’s trying to tell us something](https://reader035.vdocuments.mx/reader035/viewer/2022062314/56813432550346895d9b2162/html5/thumbnails/37.jpg)
37/49
Cool Pyramid
![Page 38: Hermite , Laguerre, Jacobi Listen to Random Matrix Theory It’s trying to tell us something](https://reader035.vdocuments.mx/reader035/viewer/2022062314/56813432550346895d9b2162/html5/thumbnails/38.jpg)
38/49
Cool Pyramid
![Page 39: Hermite , Laguerre, Jacobi Listen to Random Matrix Theory It’s trying to tell us something](https://reader035.vdocuments.mx/reader035/viewer/2022062314/56813432550346895d9b2162/html5/thumbnails/39.jpg)
39/49
Graphs on Surfaces???Thanks to Mike LaCroix• Hermite: Maps with one Vertex Coloring
• Laguerre: Bipartite Maps with multiple Vertex Colorings
• Jacobi: We know it’s there, but don’t have it quite yet.
![Page 40: Hermite , Laguerre, Jacobi Listen to Random Matrix Theory It’s trying to tell us something](https://reader035.vdocuments.mx/reader035/viewer/2022062314/56813432550346895d9b2162/html5/thumbnails/40.jpg)
40/49
The “How I met Mike” Slide
MopsDumitriu, E, Shuman 2007a=2/β
![Page 41: Hermite , Laguerre, Jacobi Listen to Random Matrix Theory It’s trying to tell us something](https://reader035.vdocuments.mx/reader035/viewer/2022062314/56813432550346895d9b2162/html5/thumbnails/41.jpg)
41/49
Multivariate Hermite and Laguerre Moments
α=2/β=1+b
![Page 42: Hermite , Laguerre, Jacobi Listen to Random Matrix Theory It’s trying to tell us something](https://reader035.vdocuments.mx/reader035/viewer/2022062314/56813432550346895d9b2162/html5/thumbnails/42.jpg)
42/49
Law S-transform
Hermite Semicircle Law
Laguerre Marcenko-Pastur Law
Jacobi Wachter Law
![Page 43: Hermite , Laguerre, Jacobi Listen to Random Matrix Theory It’s trying to tell us something](https://reader035.vdocuments.mx/reader035/viewer/2022062314/56813432550346895d9b2162/html5/thumbnails/43.jpg)
43/49
Polynomials of matrix argument
Praveen and E (2014)
Young Lattice GeneralizesSym tridiagonal
Always for β=2Only HLJ for other β?
Schur :: Jack as :: General β
β=2
![Page 44: Hermite , Laguerre, Jacobi Listen to Random Matrix Theory It’s trying to tell us something](https://reader035.vdocuments.mx/reader035/viewer/2022062314/56813432550346895d9b2162/html5/thumbnails/44.jpg)
44/49
Real, Complex, Quaternionis NOT Hermite, Laguerre,Jacobi
• We now understand that Dyson’s fascination with the three division rings lead us astray
• There is a continuum that includes β=1,2,4• Informal method, called ghosts and shadows for
β- ensembles
E. (2010)
![Page 45: Hermite , Laguerre, Jacobi Listen to Random Matrix Theory It’s trying to tell us something](https://reader035.vdocuments.mx/reader035/viewer/2022062314/56813432550346895d9b2162/html5/thumbnails/45.jpg)
45/49
Finite Random Matrix Models
Hermite Laguerre
Jacobi
![Page 46: Hermite , Laguerre, Jacobi Listen to Random Matrix Theory It’s trying to tell us something](https://reader035.vdocuments.mx/reader035/viewer/2022062314/56813432550346895d9b2162/html5/thumbnails/46.jpg)
46/49
But ghosts lead to corner’s process algorithms for H,L,J!(see Borodin, Gorin 2013)
• Hermite: Symmetric Arrow Matrix Algorithm• Laguerre: Broken Arrow Matrix Algorithm• Jacobi: Two Broken Arrow Matrices Algorithm
Dubbs, E. (2013) andDubbs, E., Praveen (2013)Also Forrester, etc.
![Page 47: Hermite , Laguerre, Jacobi Listen to Random Matrix Theory It’s trying to tell us something](https://reader035.vdocuments.mx/reader035/viewer/2022062314/56813432550346895d9b2162/html5/thumbnails/47.jpg)
47/49
Sine Kernel, Airy Kernel, Bessel Kernel NOTHermite, Laguerre, Jacobi
Bulk
Soft Edge (free)Hard Edge (fixed) Bulk
![Page 48: Hermite , Laguerre, Jacobi Listen to Random Matrix Theory It’s trying to tell us something](https://reader035.vdocuments.mx/reader035/viewer/2022062314/56813432550346895d9b2162/html5/thumbnails/48.jpg)
48/49
Conclusion
?Is there a theory???• Whose answer is
• 1) exactly Hermite, Laguerre, Jacobi• 2) or includes HLJ
• For Laguerre and Jacobi• includes all parameters
• Connects to a Matrix Framework?• Can be connected to Random Matrix Theory• Can Circle back to various
differential,difference,hypergeometric,umbral definitions?• Makes me happy!
![Page 49: Hermite , Laguerre, Jacobi Listen to Random Matrix Theory It’s trying to tell us something](https://reader035.vdocuments.mx/reader035/viewer/2022062314/56813432550346895d9b2162/html5/thumbnails/49.jpg)
49/49
Challenges for you
• In your own research: Find the hidden triad!!