Re: random matrix eigenvalues

Michael Kleder wrote:


Anyone heard of an explanation for this?

While tinkering, I noticed that a matrix that is made up of real,
*normally* distributed random entries will have eigenvalues that
seem
to be *uniformly* distributed over a disk in the complex plane. As
if
that rule isn't curious enough, occasional leakers sneak a bit
outside the unit circle, creating curious exceptions. Here's a
demo:

for n=100:500
plot(eig(randn(n))/sqrt(n),'b.')
hold on
plot(cos(0:pi/20:2*pi),sin(0:pi/20:2*pi),'r-')
hold off
axis equal
xlim([-2 2])
ylim([-2 2])
title(['scaled eigenvalues of random matrix of rank '
num2str(n)])
grid on
drawnow
end

- Mike

Gershgorin disks give you the reason why it
works, and why there are sometimes slight
excursions beyond the unit circle. At least
this explains the unit circle "bound". I'm
not sure offhand as to why the eigenvalues
appear to be uniformly distributed. There
should be a good reason for that too.

Note that the same thing holds for UNIFORM
random matrices, if you scale the eigenvalues
slightly differently. Change only one line
in your code:

plot(sqrt(12)*eig(rand(n))/sqrt(n),'b.')

HTH,
John
.

Relevant Pages

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