《无限随机矩阵理论》课程书籍文献（英文版）Handout 3 Tuesday, September

18.338J/16.394: The Mathematics of Infinite Random Matrices Experiments with the Classical Ensembles Professor Alan Edelman Handout #3, Tuesday, September 14, 2004 1 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 3. 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 nN random matrix with independent, zero mean, unit variance elements. The wigner matrix w can be obtained from G as simply G+G W 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 [4 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 intimately connected to the finite theory

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 [3]. 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 [4] 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 wigner(n, isreal) ZWIGNER The Wigner matrix WIGNER (N, ISREAL) generates an N x N symetric Wigner matrix If Isreal 1 then the elements of w are real %%%%%%%%%%%%% If IsreaL =o then the elements of w are complex N is an integer while IsrEAL equals either o or 1 w is an N x N matrix References [1] Alan Edelman, Handout 3: Experiments with Classical atrix Ensembles, Fall 2004, Course Notes 18. 338 [2] Alan Edelman, Random Matrix Eigenvalues [3] E. P. Wigner, Characteristic vectors of bordered matrices with infinite dimensions. Annals of Mathematics vo1.62, pages548-564,1955 Alan Edelman and Raj Rao, Sept. 2004 Revision:1.0$$Date:2004/09/1023:21:18$ if(isreal==1) g= randn(n, n) g =(randn(n, n)+ i*randn(n, n))/sqrt(2) g,)/ sqrt (2*n); Code 1 The Wishart Matrix (or Laguerre Ensemble) The Wishart matrices 5, also sometimes referred to as Grammian matrices 6 or sample covariance matrices, are well studied in statistics and in the book by Muirhead 7 Let g be a N x M random matrix with independent, zero mean, unit variance elements. The Wishart 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 G are i.i.d. with zero mean and unit variance. Associated with the Wishart matrix is the parameter c=N/M which is simply the ratio of the number of rows to the number of columns of the matrix G. We shall employ the notation w(c) to denote the generalized Wishart matrix with its associated parameter Code 2 lists a MATLAB function for producing a"pure"Wishart matrix from i.i. d. standard normals(randn The name Laguerre ensemble refers to the connection to the laguerre weight function and the associated Laguerre orthogonal polynomials that are so intimately 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: % [1] Alan Edelman, Handout 3: Experiments with Classical % Matrix Ensembles, Fall 2004, % Course Notes 18.338. % [2] Alan Edelman, Random Matrix Eigenvalues. % [3] 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 [5], also sometimes referred to as Grammian matrices [6] or sample covariance matrices, are well studied in statistics and in the book by Muirhead [7]. 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 = G G . (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 G are i.i.d. with zero mean and unit variance. Associated with the Wishart matrix is the parameter c = N/M which is simply the ratio of the number of rows to the number of columns of the matrix G. We shall employ the notation W(c) to denote the generalized Wishart matrix with its associated parameter c. Code 2 lists a MATLAB function for producing a “pure” Wishart matrix from i.i.d. standard normals (randn in MATLAB notation). The name Laguerre ensemble refers to the connection to the Laguerre weight function and the associated Laguerre orthogonal polynomials that are so intimately connected to the finite theory

function w= wishart(n, m, isreal) ZWISHART The Wishart matrix l WISHART(N, M, ISREAL) generate n x N Hermitian wishart matrix 1 then the elements of w are If IsreaL =0 then the elements of w are complex % are integers while IsreaL equals w is an N References l [1] Alan Edelman, Handout 3: Experiments with Classical Matrix Ensembles. Fall 2004 Course Notes 18.338 i [2] Alan Edelman, Random Matrix Eigenvalues i [3] J. Wishart, The generalized product moment distribution in samples from a normal multivariate population Biometrika, vol. 20-A, pages 32-52, 1928 i Alan Edelman and Raj Rao, Sept. 2004 %$ Revision:1.0$$Date:2004/09/1023:45:18$ if (isreal==1) randn (n, m); else end 8=(randn(n, m)+ i*randn(n, m))/sqrt(2) (g*g3) The MANoVa Matrix (or Jacobi Ensemble) The name Jacobi ensemble refers to the connection to the Jacobi weight function and the associated Jacob orthogonal polynomials that are so intimately connected to the finite theory The MANOVA matrix, can be defined in terms of two Wishart matrices, W(c1)and W(c2), as simply J=(W(c1)+W(c2)-1w(c1) The term Jacobi ensemble requires that the Wishart matrices, W(c1)and W(c2), be "pure". We shall employ the notation J(c1, c2)to denote the MANOVA matrix where c1= N/Mi and c2= N/M2 are the associated parameters of the two Wishart matrices, respectively. Note that the definition of the manova matrix implies that C1= N/MN

� � � � function w = wishart(n,m,isreal); %WISHART The Wishart matrix. % WISHART(N,M,ISREAL) generates an N x N Hermitian Wishart matrix. % If ISREAL = 1 then the elements of W are real. % If ISREAL = 0 then the elements of W are complex. % % N and M are integers while ISREAL equals either 0 or 1. % W is an N x N matrix. % % References: % [1] Alan Edelman, Handout 3: Experiments with Classical % Matrix Ensembles, Fall 2004, % Course Notes 18.338. % [2] Alan Edelman, Random Matrix Eigenvalues. % [3] J. Wishart, The generalized product moment distribution in % samples from a normal multivariate population, % Biometrika, vol. 20-A, pages 32-52, 1928. % % Alan Edelman and Raj Rao, Sept. 2004. % $Revision: 1.0 $ $Date: 2004/09/10 23:45:18 $ if(isreal==1) g = randn(n,m); else g = (randn(n,m) + i*randn(n,m))/sqrt(2); end w = (g * g’) / m; Code 2 The MANOVA Matrix (or Jacobi Ensemble) The name Jacobi ensemble refers to the connection to the Jacobi weight function and the associated Jacobi orthogonal polynomials that are so intimately connected to the finite theory. The MANOVA matrix, J, can be defined in terms of two Wishart matrices, W(c1) and W(c2), as simply J = (W(c1) + W(c2))−1 W(c1). (3) The term Jacobi ensemble requires that the Wishart matrices, W(c1) and W(c2), be “pure”. We shall employ the notation J(c1, c2) to denote the MANOVA matrix where c1 = N/M1 and c2 = N/M2 are the associated parameters of the two Wishart matrices, respectively. Note that the definition of the MANOVA matrix implies that c1 = N/M1 N

function manova(n, m1, m2, isreal) ZMANOVA The MANOVA matrix MANOVA(N, M1, M2, ISREAL) generates an N x N MANOVA matrix If Isreal 1 then the elements of w are real %%%%%%%%%%%%%% If ISREAL =0 then the elements of w are complex and M2 are integers while ISrEAL equals either o or 1 n matrix References [1] Alan Edelman, Handout assical Matrix Ensembles. Fall 2004 Course Notes 18.338 [2] Alan Edelman Random Matrix Eigenvalues [3] R. J. Muirhead, Aspects of Multivariate StatisticalTheory John Wiley Sons, New York, 1982 4%%吐 Alan Edelman and Raj Rao, Sept. 2004 SRevisi 1.0$$Dat 004/09/1023:55:18$ wishart(n, ml, isreal); Generate Wishart Matrices w2 wishart(n, m2, isreal) (w1 Code 3 Code 3 lists a MATLAB function for producing a MANOVA matrix from pure Wishart matrices

� � � � function j = manova(n,m1,m2,isreal); %MANOVA The MANOVA matrix. % MANOVA(N,M1,M2,ISREAL) generates an N x N MANOVA matrix. % If ISREAL = 1 then the elements of W are real. % If ISREAL = 0 then the elements of W are complex. % % N , M1 , and M2 are integers while ISREAL equals either 0 or 1. % N < M1. % J is an N x N matrix. % % References: % [1] Alan Edelman, Handout 3: Experiments with Classical % Matrix Ensembles, Fall 2004, % Course Notes 18.338. % [2] Alan Edelman, Random Matrix Eigenvalues. % [3] R. J. Muirhead, Aspects of Multivariate Statistical Theory, % John Wiley & Sons, New York, 1982. % % % Alan Edelman and Raj Rao, Sept. 2004. % $Revision: 1.0 $ $Date: 2004/09/10 23:55:18 $ w1 = wishart(n,m1,isreal); % Generate Wishart Matrices w2 = wishart(n,m2,isreal); j = (w1 + w2) w1; Code 3 Code 3 lists a MATLAB function for producing a MANOVA matrix from pure Wishart matrices

2 Limiting densities The limiting density of the eigenvalues of a Wigner matrix 1, 2 is given by √4-x2 De ensit The limiting density of the eigenvalues of a Wishart matrix [8 is given by 1--)6(x) where b+=(1+ va2 and I is the indicator function The limiting density of the eigenvalues of a MANOVA matrix is given by (1-2)-1+(n心 where co(r)=(CII+Ic1-2C1I2-C22+c2r2)and b+ is given by (1-c1)2 ci-c1+2+e-c1c2平2√a1+e2-c1c2 Verify that the MATLAB functions listed do indeed compute these densities symbolically function f wigneredf ZWIGNEREDF Density of eigenvalues of an infinite Wigner matrix WIGNEREDF returns the theoretical density of the eigenvalues of an infinite Wigner matrix %%%%%% F is a bolic variable References [1] Alan Edelman, Handout 3: Experiments with Classical % latrix Ensembles. Fall Course Notes 18.338 [2] Alan Edelman, Random Matrix Eigenvalues [3] E. P. Wigner, Characteristic vectors of bordered matrices with % infinite dimensions, Annals of Mathematics, % o1.62, pages548-564,1955 Alan Edelman and Ra] Rao, Sept. 2004 Revision:1.0$$Date:2004/09/1120:47:15$ f=sqrt(4-x^2)/(2*p1); Code 4

� � � � � � � � � � � � 2 Limiting densities The limiting density of the eigenvalues of a Wigner matrix [1, 2] is given by √ 4 − x2 Density = . (4) 2 π The limiting density of the eigenvalues of a Wishart matrix [8] is given by 1 Density = max 0, 1 − δ(x) + (x − b−)(b+ − x) I[ b−,b+] (5) c 2πxc where b± = (1 ± √c)2 and I is the indicator function. The limiting density of the eigenvalues of a MANOVA matrix is given by Density = max 0, 1 − 1 δ(x − 1) + (x − b−)(b+ − x) I[ b−,b+] (6) c2 2πc0(x) 2 3 where c0(x) = c1 x + x3c1 − 2 c1 x − c2 x + c2 x2 and b± is given by b± = 2 (1 − c1)2 . (7) c1 − c1 + 2 + c2 − c1c2 ∓ 2 √c1 + c2 − c1c2 Verify that the MATLAB functions listed do indeed compute these densities symbolically. function f = wigneredf; %WIGNEREDF Density of eigenvalues of an infinite Wigner matrix. % WIGNEREDF returns the theoretical density of the eigenvalues of % an infinite Wigner matrix % % F is a symbolic variable. % % References: % [1] Alan Edelman, Handout 3: Experiments with Classical % Matrix Ensembles, Fall 2004, % Course Notes 18.338. % [2] Alan Edelman, Random Matrix Eigenvalues. % [3] 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: 2004/09/11 20:47:15 $ syms f x f = sqrt(4-x^2)/(2*pi); Code 4

function f wishartedf(c) ZWISHARTEDF Density of eigenvalues of an infinite Wishart matrix THWISHART(C) returns the theoretical limiting density of the eigenvalues of an infinite Wishart matrix with c =N/ M where N and M are the parameters of the wishart %%%%%%%%%%%%%%%%%%%s positive real number less than 1 F is a symbolic variable References [1] Alan Edelman, Handout 3: Experiments with Classical Matrix Ensembles. Fall 2004 Course Notes 18. 338 [2] Alan Edelman, Random Matrix Eigenvalues [3]J. Wishart, The generalized product moment distribution in samples from a normal multivariate population Biometrika, vol. 20-A, pages 32-52, 1928 Alan Ede lme d Raj Rao, Sept. 2004 SRevisio $$Date:2004/09/1120:49:20$ f (1-sqrt(c))-2 (1+sqrt(c))-2; f=sqrt((x-a1)*(a2-x))/(2*pi*x*C); Code 5

� � � � function f = wishartedf(c) %WISHARTEDF Density of eigenvalues of an infinite Wishart matrix. % THWISHART(C) returns the theoretical limiting density of the % eigenvalues of an infinite Wishart matrix with % C = N / M where N and M are the parameters of % the Wishart matrix. % % C is a positive real number less than 1. % F is a symbolic variable. % % References: % [1] Alan Edelman, Handout 3: Experiments with Classical % Matrix Ensembles, Fall 2004, % Course Notes 18.338. % [2] Alan Edelman, Random Matrix Eigenvalues. % [3] J. Wishart, The generalized product moment distribution in % samples from a normal multivariate population, % Biometrika, vol. 20-A, pages 32-52, 1928. % % Alan Edelman and Raj Rao, Sept. 2004. % $Revision: 1.0 $ $Date: 2004/09/11 20:49:20 $ syms f x a1 = (1-sqrt(c))^2; a2 = (1+sqrt(c))^2; f = sqrt((x-a1)*(a2-x))/(2*pi*x*c); Code 5

function f manovaedf(c1, c2) /MANOVAEDF Theoretical density of eigenvalues of an infinite MANOVA matrix MANOV AEDF(C1, C2) returns the limiting theoretical density of the eigenvalues of an infinite MANOVA matrix %%%%%%%%%%%% where C1 M1 M2 are parameters of the MANOVA matr C1 and C2 are positive real numbers less than F symbolic variabl References [1] Alan Edelman, Handout 3: Experiments with Classical Matrix Ensembles. Fall 2004 Course Notes 18.338 [2] Alan Edelman Random Matrix Eigenvalues [3] R. J. Muirhead, Aspects of Multivariate StatisticalTheory John Wiley Sons, New York, 1982 %%%s Alan Edelman and Ra] Rao, Sept. 2004 SRevisi 1.0$$Dat 004/09/1023:55:18$ 2: *x-2 f sqrt(4*b2*b0-b1^2)/(2*pi*b2); Code 6 3 Experiments Put all code in the same directory and run the script classicexpt. Try it with different sizes of matrices and see how quickly the histograms converge

� � � � 3 function f = manovaedf(c1,c2); %MANOVAEDF Theoretical density of eigenvalues of an infinite MANOVA matrix. % MANOVAEDF(C1,C2) returns the limiting theoretical density of the % eigenvalues of an infinite MANOVA matrix, % where C1 = N / M1 and C2 = N / M2 and N1, M1, and % M2 are parameters of the MANOVA matrix. % % C1 and C2 are positive real numbers less than 1. % F is a symbolic variable. % % References: % [1] Alan Edelman, Handout 3: Experiments with Classical % Matrix Ensembles, Fall 2004, % Course Notes 18.338. % [2] Alan Edelman, Random Matrix Eigenvalues. % [3] R. J. Muirhead, Aspects of Multivariate Statistical Theory, % John Wiley & Sons, New York, 1982. % % % Alan Edelman and Raj Rao, Sept. 2004. % $Revision: 1.0 $ $Date: 2004/09/10 23:55:18 $ syms x f b0 = c1*x-c2*x-c1+2; b1 = -2*c2*x^2+2*x-3*c1*x+c1+c2*x-1+2*c1*x^2; b2 = c1*x-2*c1*x^2+c2*x^2-x^3*c2+x^3*c1; f = sqrt(4*b2*b0-b1^2)/(2*pi*b2); Code 6 Experiments . Put all in the same directory and run the script classicexpt. Try it with different sizes of matrices and see how quickly the histograms converge. code

c⊥ assicexpt.m 7Code 1.2 of Random Eigenvalues by Alan Edelman and Raj Rao 7Experiment: Generate many realizations of the classical r om matri ces oBservation: Histogram the eigenvalues of the random matrices /Theory Falls on the predicted curves REference [1] Alan Edelman, Handout 3, Experiments with Classical % Matrix Ensembles, Fall 2004 [2 Alan Edelman, Random Matrix Eigenvalues trials = 1000; y Number of trials l Array for collecting eigenvalues l Symbolic variables for theoretical predictions l Requires Symbolic Toolbox in MATLAB figure; set(gcf, ' DoubleBuffer', 'On') warnin for i trials i Generate random matrix (uncomment the appropriate line isreal =1 %N=50;C wigner(N, isreal); f wigneredf N=50: M=100: C wishart (N,M,isreal): f= wishartedf (N/M) A N=50: M1=100; M2=100; C manova(N, M1, M2, isreal);f manovaedf(N/M1, NM2 i Collect the eigenvalues e [e; real(eig(C))]: Take real part to compensate for numerical isspes h Plot density of eigenvalues x0=-2: binsize=0. 05: xf=4 Th, hn, span]= hist(e, O, binsize, xf); axis(Lxo xf 0 1.5]): hold on Plot theory F real(subs(f, span)): F(F=inf)=0; hold on ts=['# trials num2str(i)]; title(ts, FontSize, 15) plot (span, F,'red','LineWidth, 2); drawnow; hold off end Code 7

� � � � %classicexpt.m %Code 1.2 of Random Eigenvalues by Alan Edelman and Raj Rao %Experiment: Generate many realizations of the classical random matrices. %Observation: Histogram the eigenvalues of the random matrices. %Theory: Falls on the predicted curves. %Reference: [1] Alan Edelman, Handout 3, Experiments with Classical % Matrix Ensembles, Fall 2004. % [2] Alan Edelman, Random Matrix Eigenvalues. trials = 1000; % Number of trials e = []; % Array for collecting eigenvalues syms f x; % Symbolic variables for theoretical predictions % Requires Symbolic Toolbox in MATLAB figure; set(gcf,’DoubleBuffer’,’On’); warning off; for i = 1 : trials % Generate random matrix (uncomment the appropriate line) isreal = 1; % N=50; C = wigner(N,isreal); f = wigneredf; % N=50; M=100; C = wishart(N,M,isreal); f = wishartedf(N/M); % N=50; M1=100; M2=100; C = manova(N,M1,M2,isreal); f = manovaedf(N/M1,N/M2); % Collect the eigenvalues e = [e;real(eig(C))]; % Take real part to compensate for numerical issues % Plot density of eigenvalues x0=-2; binsize=0.05; xf=4; [h,hn,xspan] = histn(e,x0,binsize,xf); axis([x0 xf 0 1.5]); hold on % Plot theory F = real(subs(f,xspan)); F(F==inf)=0; hold on ts=[’# trials = ’ num2str(i)]; title(ts,’FontSize’,15); plot(xspan,F,’red’,’LineWidth’,2);drawnow; hold off end Code 7

4 Exercises Try the following by appropriately modifying the scripts /functions described in this handout Do the Gaussian matrices have to be circular for the limiting density expressions to be valid? When isreal=let the real and the complex parts have different variances (keep the sum of the variances equal to 11). What happens? Try other distributions such as sign (randn), the exponential distribution. What happens? Try a distribution with non-zero mean. What can you say about the limiting density? Do the number of trials or the size of the matrix at which the density starts converging " depend on the distribution? Or on whether it is real or complex? Compute the first k moments of these ensembles. Plot the mean and the variance of these moment as a function of the matrix size m References [1 Eugene P. Wigner. Characteristic vectors of bordered matrices with infinite dimensions. Annals of Math,62548-564,1955 2 Eugene P. Wigner. On the distribution of the roots of certain symmetric matrices. Annals of Mathematics, 67: 325-327. 1958. 3 Madan Lal Mehta. Random Matrices Academic Press, Boston, second edition, 1991 [4 Freeman J. Dyson. A class of matrix ensembles. J. Math. Phys. 13: 90-97, 1972 5 J. Wishart. The generalized product moment distribution in samples from a normal multivariate population. Biometrika, 20 A: 32-52, 1928 [6 V L Girko. Spectral Theory of Random Matrices (RussiaNauka, Moscow, 1988 [7 Robb J Muirhead. Aspects of Multivariate Statistical Theorlphn Wiley &z Sons, New York, 1982 18 V.A. Marcenko and L A. Pastur. Distribution of eigenvalues for some sets of random matrices. Math USSR Cornil1:457-483,1967

4 Exercises Try the following by appropriately modifying the scripts/functions described in this handout: • Do the Gaussian matrices have to be circular for the limiting density expressions to be valid? When isreal = 0 let the real and the complex parts have different variances (keep the sum of the variances equal to 1!). What happens? • Try other distributions such as sign(randn), the exponential distribution. What happens? • Try a distribution with non-zero mean. What can you say about the limiting density? • Do the number of trials or the size of the matrix at which the density “starts converging” depend on the distribution? Or on whether it is real or complex? • Compute the first k moments of these ensembles. Plot the mean and the variance of these moments as a function of the matrix size N. References [1] Eugene P. Wigner. Characteristic vectors of bordered matrices with infinite dimensions. Annals of Math., 62:548–564, 1955. [2] Eugene P. Wigner. On the distribution of the roots of certain symmetric matrices. Annals of Mathematics, 67:325–327, 1958. [3] Madan Lal Mehta. Random Matrices. Academic Press, Boston, second edition, 1991. [4] Freeman J. Dyson. A class of matrix ensembles. J. Math. Phys., 13:90–97, 1972. [5] J. Wishart. The generalized product moment distribution in samples from a normal multivariate population. Biometrika, 20 A:32–52, 1928. [6] V. L. Girko. Spectral Theory of Random Matrices (Russian) . Nauka, Moscow, 1988. [7] Robb J. Muirhead. Aspects of Multivariate Statistical Theory . John Wiley & Sons, New York, 1982. [8] V.A. Marˇcenko and L.A. Pastur. Distribution of eigenvalues for some sets of random matrices. Math USSR Sbornik, 1:457–483, 1967