Re: Small random matrices

From: Herman Rubin (hrubin_at_stat.purdue.edu)
Date: 09/25/04 

Date: Sat, 25 Sep 2004 17:30:07 +0000 (UTC)

In article <ciuj4r$4jq$1@news.ks.uiuc.edu>,
Nicolas Le Bihan <Nicolas.Le-Bihan@lis.inpg.fr> wrote:

>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 ?

They cannot be, as a determines a*. If you assume joint
normality, the elements have a jointly normal distribution.
If by M you mean the determinant, this is |a|^2 + |b|^2,
and the moment generating function of this is not at all
difficult to obtain, again on joint normality. From the
marginals, the question cannot be answered, except for the
expectation of M. 

-- 
This address is for information only.  I do not claim that these views
are those of the Statistics Department or of Purdue University.
Herman Rubin, Department of Statistics, Purdue University
hrubin@stat.purdue.edu         Phone: (765)494-6054   FAX: (765)494-0558

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