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