Re: bi-Unitarily invariant matrix
- From: hrubin@xxxxxxxxxxxxxxxxxxxx (Herman Rubin)
- Date: 28 Feb 2007 21:25:40 -0500
In article <29797763.1172660247398.JavaMail.jakarta@xxxxxxxxxxxxxxxxxxxxxx>,
Kammoun <kammoun@xxxxxxx> wrote:
Hi,
I have often read that a standard random gaussian matrix is bi unitarily invariant, ie if we denote by U a unitary random matrix, X a standard gaussian matrix independent of U than UX has the same statistics as X.
This result is often cited in articles but I haven't found a proof of it. Have you any idea about the proof?
Thank you.
The simplest way of seeing this is that the density function
of a standard multivariate normal distribution (or more
generally, independent with the same variance) is symmetric
under orthogonal transformations. So as long as a unitary U
is independent of a standard normal vector y, Uy has the
same distribution as y by symmetry.
--
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@xxxxxxxxxxxxxxx Phone: (765)494-6054 FAX: (765)494-0558
.
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