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handout5 - 18.338J/16.394J: The Mathematics of Infinite...

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Unformatted text preview: 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 n , n = 0 , 1 , 2 , . . . , satisfying integraldisplay b a m n w ( x ) dx = mn . (1) The polynomials can be generated easily, because they satisfy a three-term recurrence ( x ) = parenleftBig integraltext b a w parenrightBig- 1 / 2 , (2) x n ( x ) = b n n- 1 ( x ) + a n +1 n ( x ) + b n +1 n +1 ( x ) . (3) (Note that b is taken to be zero.) Perhaps surprisingly, the three-term recurrence can be used to find the roots of n as well. Simply form the symmetric tridiagonal matrix T n = a 1 b 1 b 1 a 2 b 2 b 2 a 3 b 3 . . . ....
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handout5 - 18.338J/16.394J: The Mathematics of Infinite...

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