Re: Small random matrices

From: Nicolas Le Bihan (Nicolas.Le-Bihan_at_lis.inpg.fr)
Date: 09/23/04 

Date: Thu, 23 Sep 2004 13:30:03 +0000 (UTC)

Thanks.
I must give more details to be clearer..:)

In fact, I am looking for a formulation of the pdf of any 2x2 matrix M
of the form :

  | a b |
M=| |
  | -b* a*|

where a and b are complex random variables and * is the conjugation.
Such matrices are normal. So, given the marginal distributions of
a,a*,b and b*, I want the pdf of M (a,a*,b and b* may not be
independent).

Now, the second part of my question is, can I define some moments and
cumulants of M ? And how ?

Thanks again Robert.
Nicolas

On Wed, 22 Sep 2004 21:30:02 +0000 (UTC), Robert Israel wrote:
>
>I'm not sure I understand your question. The pdf of the matrix A
_is_ the
>joint pdf of the matrix elements. Of course, you have to pick that
pdf
>carefully to make sure the matrix is normal - in particular, except
in
>trivial cases you won't be able to make the elements independent.
>Perhaps what you are asking is, given (marginal) pdf's for the
>matrix elements, can you find a pdf for the matrix A that gives these
>marginal pdf's and makes A normal? The answer to that is "maybe":
>there are certainly constraints on the marginal pdf's. When it
>is possible, I think it would be rather complicated, even in the
>2 x 2 case.
>
>Robert Israel israel@math.ubc.ca
>Department of Mathematics <a
href="http://www.math.ubc.ca/~israel">http://www.math.ubc.ca/~israel>

>University of British Columbia Vancouver, BC, Canada