18.338J/16.394J: The Mathematics of Infinite Random Matrices Tridiagonal Matrices, Orthogonal Polynomials and the Classical Random matrix ensemble Brian sutton Handout #5, Thursday, September 23, 2004 In class. we saw the connection between the so-called Hermite matrix and the semi-circular law. There is actually a deeper story that connects the classical random matrix ensembles to the classical orthogonal polynomials studied in classical texts such as [1 and more recent monographs such as 2. We illuminate partofthisstoryhereThewebsitewww.mathworld.comisanexcellentreferenceforthesepolynomialsand will prove handy when completing the exercises In any computational explorations, see if you can spot the interesting feature in the eigenvectors(either the first or last row/ column) of the corresponding tridiagonal matrix. 1 Roots of orthogonal polynomials Any weight function w ()on an interval a, b(possibly infinite) defines a system of orthogonal polynomials satisfying (a)dx The polynomials can be generated easily, because they satisfy a three-term recurrence n()=() (x)=bn丌n-1(x)+an+1丌n(x)+bn+1丌n+1( (Note that bo is taken to be zero. Perhaps surprisingly, the three-term recurrence can be used to find the roots of Tn as well. Simply form the symmetric tridiagonal matrix b1 b3 The roots of Tn are the eigenvalues of Tn! Remark 1. The symmetry of T follows from two choices which we made. First, in(1), Tn was normalized to have unit length under the inner product(f, g)=fgw. Second, (3)was arranged in the form ITn= RHS Not all authors follow these conventions
18.338J/16.394J: The Mathematics of Infinite Random Matrices Tridiagonal Matrices, Orthogonal Polynomials and the Classical Random Matrix Ensembles Brian Sutton Handout #5, Thursday, September 23, 2004 In class, we saw the connection between the so-called Hermite matrix and the semi-circular law. There is actually a deeper story that connects the classical random matrix ensembles to the classical orthogonal polynomials studied in classical texts such as [1] and more recent monographs such as [2]. We illuminate part of this story here. The website www.mathworld.com is an excellent reference for these polynomials and will prove handy when completing the exercises. In any computational explorations, see if you can spot the interesting feature in the eigenvectors (either the first or last row/column) of the corresponding tridiagonal matrix. 1 Roots of orthogonal polynomials Any weight function w(x) on an interval [a, b] (possibly infinite) defines a system of orthogonal polynomials πn, n = 0, 1, 2, . . . , satisfying Z b a πmπnw(x) dx = δmn. (1) The polynomials can be generated easily, because they satisfy a three-term recurrence π0(x) = R b a w −1/2 , (2) xπn(x) = bnπn−1(x) + an+1πn(x) + bn+1πn+1(x). (3) (Note that b0 is taken to be zero.) Perhaps surprisingly, the three-term recurrence can be used to find the roots of πn as well. Simply form the symmetric tridiagonal matrix Tn = a1 b1 b1 a2 b2 b2 a3 b3 . . . . . . . . . bn−2 an−1 bn−1 bn−1 an . The roots of πn are the eigenvalues of Tn! Remark 1. The symmetry of T follows from two choices which we made. First, in (1), πn was normalized to have unit length under the inner product hf, gi = R b a fgw. Second, (3) was arranged in the form xπn = RHS. Not all authors follow these conventions
To see that the eigenvalues are the roots, consider vectors of the form U=(x0(x),r1(x),…,丌n-1(x)) The recurrence(3)implies that(Tu)i=TTi-la)=avi for i n-1, regardless of the value of Now, if a is a root of Tn(), then(3)specializes to x丌n-1(x)=bn-1丌n-2(x)+an丌n-1x TUn=bn-1Un-1+anUn =(Tu)n Therefore, every root of Tn(a)is an eigenvalue of Tn 2 Hermite polynomials Let w(a)=e- on R. According to mathworld. wolfram. com, the system of polynomials Hn defined by Hm Hnw(a)dr=5mn2nIVT satisfies the recurrence H0(x)=1 Hn+1(r)=2c Hn(a)-2nHn-l(a (5) Let the orthonormal Hermite polynomials be defined by Hn(a)=5/2-1zzHn(a), so that and substitute 2n/(n!)1/21/AHn(a)for Hn(r)in(5)to find zHn(r)=VaVnHn-1(z)+Vn+lHn+1(r) These equations are exactly of the form (1)-3), so the eigenvalues of 10 are the roots of Hn Exericse: Write a MATLAB function that generates this tridiagonal matrix. Use the eig command to compute its eigenvalues for any choice of n. Histogram the roots of Hn for n= 50, 100, 250, 300, 500 generalized ensemble is real or complex? the semi-circular law. Does it make a difference whether the the sing the hist functio on. Compare
To see that the eigenvalues are the roots, consider vectors of the form v = (π0(x), π1(x), . . . , πn−1(x))T . The recurrence (3) implies that (T v)i = xπi−1(x) = xvi for i = 1, . . . , n − 1, regardless of the value of x. Now, if x is a root of πn(x), then (3) specializes to xπn−1(x) = bn−1πn−2(x) + anπn−1x i.e. xvn = bn−1vn−1 + anvn = (T v)n. Therefore, every root of πn(x) is an eigenvalue of Tn. 2 Hermite polynomials Let w(x) = e −x 2 on R. According to mathworld.wolfram.com, the system of polynomials H˜ n defined by Z ∞ −∞ H˜mH˜ nw(x) dx = δmn2 nn! √ π satisfies the recurrence H˜ 0(x) = 1 (4) H˜ n+1(x) = 2xH˜ n(x) − 2nH˜ n−1(x). (5) Let the orthonormal Hermite polynomials be defined by Hn(x) = 1 2n/2 √ n!π1/4 H˜ n(x), so that Z ∞ −∞ HmHnw dx = δmn, and substitute 2n/2 (n!)1/2π 1/4Hn(x) for H˜ n(x) in (5) to find H0(x) = π 1/4 xHn(x) = √ 1 2 √ nHn−1(x) + √ 1 2 √ n + 1Hn+1(x). These equations are exactly of the form (1)-(3), so the eigenvalues of 1 √ 2 0 √ 1 √ 1 0 √ 2 0 √ 2 0 √ 3 . . . . . . . . . √ n − 2 0 √ n − 1 √ n − 1 0 are the roots of Hn. Exericse: Write a MATLAB function that generates this tridiagonal matrix. Use the eig command to compute its eigenvalues for any choice of n . Histogram the roots of Hn for n = 50, 100, 250, 300, 500 using the histn function. Compare it to the semi-circular law. Does it make a difference whether the the generalized ensemble is real or complex?
3 Laguerre polynomials Let Wo(x)=x°e-on[0,∞) Erercise: Verify that the Laguerre polynomials defined by satisfy the recurrence L8(x)= rLm(x)=-Va+nLm-1(ax)+(a+1+2n)Lm(x)-vn+va+n+ILa+1(x) Therefore, the roots of Ln are given by the eigenvalues of a+1 √T√a+T 0 √2√a+2a+5 -√3√a +n-1 Exericse: Write a MATLAB function that generates this tridiagonal matrix. Use the eig command to compute its eigenvalues for any choice of n and a. Histogram the roots of Ln for different values of n and m using the hist function. Compare it to the Marcenko-Pastur distribution for the generalized laguer Ensemble. Does it make a difference whether the the generalized ensemble is real or complex? Note a=B(n-m+1)/2-l where B=l when the elements are real and B=2 when the elements are complex The parameters m and n are associated with the rows and columns of the Gaussian matrix used t construct the Wishart matrix. Recall that m/n=c 4 Jacobi polynomials Jacobi polynomials are orthogonal with respect to the weight w(x)=(l-r)(1+r)on the interval Erercise: Verify that the tridiagonal matrix describing the recurrence is produced by the following MATLaB
3 Laguerre polynomials Let wα(x) = x αe −x on [0, ∞). Exercise: Verify that the Laguerre polynomials defined by Z ∞ 0 L α mL α nw dx = δmn satisfy the recurrence L α 0 (x) = 1 p Γ(α + 1) xLα n (x) = − √ n √ α + nLα n−1 (x) + (α + 1 + 2n)L α n (x) − √ n + 1√ α + n + 1L α n+1(x). Therefore, the roots of L α n are given by the eigenvalues of α + 1 − √ 1 √ α + 1 − √ 1 √ α + 1 α + 3 − √ 2 √ α + 2 0 − √ 2 √ α + 2 α + 5 − √ 3 √ α + 3 . . . . . . . . . − √ n − 1 √ α + n − 1 α − 1 + 2n . Exericse: Write a MATLAB function that generates this tridiagonal matrix. Use the eig command to compute its eigenvalues for any choice of n and α. Histogram the roots of Ln for different values of n and m using the histn function. Compare it to the Marˇcenko-Pastur distribution for the generalized Laguerre Ensemble. Does it make a difference whether the the generalized ensemble is real or complex? Note: α = β (n − m + 1)/2 − 1 where β = 1 when the elements are real and β = 2 when the elements are complex. The parameters m and n are associated with the rows and columns of the Gaussian matrix used to construct the Wishart matrix. Recall that m/n = c. 4 Jacobi polynomials Jacobi polynomials are orthogonal with respect to the weight w(x) = (1 − x) α(1 + x) β on the interval [−1, 1]. Exercise: Verify that the tridiagonal matrix describing the recurrence is produced by the following MATLAB code
function t= trijacobi(n, a, b) lTRIJACOBI Returns the tridiagonal matrix of polynomial recurrence coefficients l TRIJACOBI(, A, B) returns a tridiagonal matrix whose entries correspond to the coefficients describing the recurrence between the Jacobi Polynomials % The eigenvalues of this matrix correspond to the roots of the N-th Jacobi polynomial with parameters A and B %%%%%%%% A and B are integers References [1] Alan Edel Handout 6: Tridiagonal Matrices, Orthogonal Polynomials ans the Classical Random Matrix Ensembles. Fall 2004 Course notes 18.338 i [2] Alan Edelman, Random Matrix Eigenvalues i [3] Gabor Szego, Orthogonal Polynomials, American Mathematical Society, Providence, 1975. 4th Edition [4] Eric Weisstein, Jacobi Polynomial. From MathWorld A Wolfram Web Resource %%%% http://mathworld.wolfram.com/jacobipolyNomial.htm Brian Sutton, Sep %$ Revision:1.0$$Date:2004/09/2311:35:18$ i=(1:n-1) d1=2*sqrt(i.*(a+1).*(b+1).*(a+b+1)./(-1+a+b+2*1)./(1+a+b+2*1)./(a+ b+2*i) d0=-(a^2-b^2)./(a+b+2*(0:n-1))./(2+a+b+2*(0:n-1)) t= spdiags([[d1; o] do [o: d1]],[-1 0 1],n,n); Code 1 Exericse: Use the eig command to compute its eigenvalues for any choice of n. Histogram the roots of Tn for different values of n, a and b using the hist function. Compare it to the Manova density for the MANOVA matrix. Hint: You might have to shift the spectrum for it to line up exactly. Does it make a difference whether the the generalized ensemble is real or complex? Note: a=B(ni-m+1)/2-1 and b=B(n2-m+1)/2-1 where B= l when the elements are real and B=2 when the elements are plex. The parameters m, n1 and n2 are associated with the rows and columns of the Gaussian matrices to construct the MANOVA matrix. Recall that m/n1= C1 and m/n2 =Cy References 1 Gabor Szego. Orthogonal Polynomials. American Mathematical Society, Providence, 1975. 4th edition 2 Percy Deift. Orthogonal Polynomials and Random Matrices: A Riemann-Hilbert Approach. American Mathematical Society, Providence, 1998
✬ ✫ ✩ ✪ function t = trijacobi(n,a,b) %TRIJACOBI Returns the tridiagonal matrix of polynomial recurrence coefficients % TRIJACOBI(N,A,B) returns a tridiagonal matrix whose entries correspond % to the coefficients describing the recurrence between % the Jacobi Polynomials. % % The eigenvalues of this matrix correspond to the roots of the N-th % Jacobi polynomial with parameters A and B. % % N, A and B are integers. % % % References: % [1] Alan Edelman, Handout 6: Tridiagonal Matrices, Orthogonal % Polynomials ans the Classical Random Matrix % Ensembles, Fall 2004, % Course Notes 18.338. % [2] Alan Edelman, Random Matrix Eigenvalues. % [3] Gabor Szego, Orthogonal Polynomials, American Mathematical % Society, Providence, 1975. 4th Edition. % [4] Eric Weisstein, Jacobi Polynomial." From MathWorld-- % A Wolfram Web Resource. % http://mathworld.wolfram.com/JacobiPolynomial.htm % % Brian Sutton, Sept. 2004. % $Revision: 1.0 $ $Date: 2004/09/23 11:35:18 $ i = (1:n-1)’; d1 = 2*sqrt(i.*(a+i).*(b+i).*(a+b+i)./(-1+a+b+2*i)./(1+a+b+2*i))./(a+ ... b+2*i); d0 = -(a^2-b^2)./(a+b+2*(0:n-1)’)./(2+a+b+2*(0:n-1)’); t = spdiags([[d1;0] d0 [0;d1]],[-1 0 1],n,n); Code 1 Exericse: Use the eig command to compute its eigenvalues for any choice of n. Histogram the roots of Tn for different values of n, a and b using the histn function. Compare it to the MANOVA density for the MANOVA matrix. Hint: You might have to shift the spectrum for it to line up exactly. Does it make a difference whether the the generalized ensemble is real or complex? Note: a = β (n1 − m + 1)/2 − 1 and b = β (n2 − m + 1)/2 − 1 where β = 1 when the elements are real and β = 2 when the elements are complex. The parameters m, n1 and n2 are associated with the rows and columns of the Gaussian matrices used to construct the MANOVA matrix. Recall that m/n1 = c1 and m/n2 = c2. References [1] G´abor Szeg¨o. Orthogonal Polynomials. American Mathematical Society, Providence, 1975. 4th edition. [2] Percy Deift. Orthogonal Polynomials and Random Matrices: A Riemann-Hilbert Approach. American Mathematical Society, Providence, 1998
