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Unformatted text preview: 1 18.338J/16.394J: The Mathematics of Infinite Random Matrices Experiments with the Classical Ensembles Professor Alan Edelman Handout #3, Tuesday, September 14, 2004 The Classical Random Matrix Ensembles The Wigner Matrix (or Hermite Ensemble) The Wigner matrices [1, 2] are often known as the Hermite or Gaussian ensembles are well studied in physics and in the book by Mehta . The term Wigner matrix does not require the entries be normal, though the term Gaussian ensemble and Hermite ensemble does. Typically what is required is that the elements have the same independence properties and variance as the Gaussian case. This means variance 1/n off the diagonal and sqrt(2)/n on the diagonal, and elements independent on the upper triangle. Let G be a N N random matrix with independent, zero mean, unit variance elements. The Wigner matrix W can be obtained from G as simply G + G W = . (1) 2 N Code 1 lists a MATLAB function for producting a Wigner matrix from i.i.d. standard normals ( randn in MATLAB notation). The specific names Gaussian orthogonal ensemble, Gaussian unitary ensemble, and Gaussian symplectic ensemble or GOE, GUE, GSE were named by Dyson  and refer to the real, complex, and quaternion cases respectively. These names refer to invariance properties of the distribution. The name Hermite ensemble refers to the connection to the Hermite weight function and the associated Hermite orthogonal polynomials that are so intimitely connected to the finite theory. function w = wigner(n,isreal); %WIGNER The Wigner matrix. % WIGNER(N,ISREAL) generates an N x N symettric Wigner matrix. % If ISREAL = 1 then the elements of W are real. % If ISREAL = 0 then the elements of W are complex. % % N is an integer while ISREAL equals either 0 or 1. % W is an N x N matrix. % % References: %  Alan Edelman, Handout 3: Experiments with Classical % Matrix Ensembles, Fall 2004, % Course Notes 18.338. %  Alan Edelman, Random Matrix Eigenvalues. %  E. P. Wigner, Characteristic vectors of bordered matrices with % infinite dimensions, Annals of Mathematics, % vol. 62, pages 548-564, 1955. % % Alan Edelman and Raj Rao, Sept. 2004. % $Revision: 1.0 $ $Date: 2004/09/10 23:21:18 $ if(isreal==1) g = randn(n,n); else g = (randn(n,n) + i*randn(n,n))/sqrt(2); end w = (g + g) / sqrt(2*n); Code 1 The Wishart Matrix (or Laguerre Ensemble) The Wishart matrices , also sometimes referred to as Grammian matrices  or sample covariance matrices, are well studied in statistics and in the book by Muirhead . Let G be a N M random matrix with independent, zero mean, unit variance elements. The Wishart matrix W can be obtained from G as simply 1 W = GG . (2) M The term Wishart matrix does require that the entries of G are i.i.d. normally distributed elements as does the term Laguerre ensemble. We shall refer to this as a pure Wishart matrix when we need to make that distinction. When analyzing the limiting density, typically what is required is that the elements of distinction....
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