Date: Jun 12, 2012 5:20 PM
Author: kumar vishwajeet
Subject: basic query about probability density function plot
Hi,

I've a very basic query about the plot of eigenvalue distribution of a random matrix.:

I've a random matrix(NxN) with i.i.d entries having a pdf 'p'. The joint density function of its eigenvalues(l_i) is defined as,

K*exp(-0.5*sum(l_i^2))*[product(i to N-1){1/i}*product(j>i to N){(l_i-l_j)^2)}

My query is how to plot its pdf??? Because for one matrix, we'll get only a point value of joint pdf of the eigenvalues.

Let us assume that N=2, so that we do not have to marginalize for plotting.

My understanding:

1. Make m=1000 samples of random matrix. Each sample of size NxN = 2x2.

2. Thus we get 1000 sets of eigenvalues. Each set has 2 eigenvalues, l_ai, l_bi. where i denotes the set number.

3. Make a meshgrid of X = [l_a1 l_a2 l_a3.....l_a1000] and Y = [l_b1 l_b2 l_b3....l_b1000].

4. Find the joint pdf,p_l, of X and Y using the formula written above.

5. surf(X,Y,p_l).

Is it correct??? Or, Am I making some mistake??