mardi 26 avril 2016

What shape should we get by plotting the log-log graph of Inverse Performance Ratio of eigenvectors vs Eigenvalues for a random matrix?

The Inverse Participation Ratio (I.P.R.) of a vector u = (u1, .... um) for i = 1, ..., m is defined as follows:

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When plotting the log-log of IPR of the eigenvectors vs the eigenvalues, L, we should get something interesting, perhaps a straight line like:

enter image description here

But I am getting a haphazard thing:

enter image description here

This is my code.

m=98; n=753;
H=randn(m,n);
W=1/n*(H*(H'));
[U, lambda] = eig(W);

for i=1:size(U,2)
    IPR(i,1)=0;
    for j=1:98
        IPR(i,1)=IPR(i,1)+U(j,i)^4;
    end
    L(i,1)=lambda(i,i);
end

loglog(L,IPR);

Could anyone please point out what I am doing wrong?




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