Question

3. Random matrices are matrix generalization of random variables. To examine the distribution of eigenvalues of such matrices, first generate a 100x100 random matrix with elements selected from a random distribution of ones and zeros (to do this you can start with the function rand( ) together with round( ) or alternatively randi( )) and then find the absolute value (the norm) of each the eigenvalues of this matrix with the routine eig。 Repeat this 100 times storing all the absolute values in a single 10,000 element row vector This can be performed in a very simple manner by noting that writing e.g. aR(1)-4; aR(2) -3; creates a two component row vector with values [4, 3]. Finally, with lookfor find the Octave function that generates a histogram from a set of values and then employ help to determine how to incorporate this function into your program so that you generate a histogram with 200 bins for the result (the answer may appear a bit unusual). Repeat this procedure for Gaussian distributed random variables with zero mean value and unit variance (substitute randn() for rand() and eliminate the round() function). Employ the subplot() routine to plot both the uniform and Gaussian random variable results side by side on the same page

Answer 1

Octave Script:

close all
clear
clc

vec = [];
for i = 1:100
x = randi(1000, 100);
eig_vals = abs(eig(x));
eig_vals = eig_vals(:)';
vec = [vec eig_vals];
end
subplot(211), hist(vec, 200), title('Uniform Distribution')

vec = [];
for i = 1:100
x = randn(100);
eig_vals = abs(eig(x));
eig_vals = eig_vals(:)';
vec = [vec eig_vals];
end
subplot(212), hist(vec, 200), title('Gaussian Distribution')

Output:

Uniform Distribution 2000 1500 1000 500 60000 50000 40000 30000 20000 10000 0 Gaussian Distribution 140 120 100 80 60 40 20 12 10 6 4 2