From: kovarik@mcmail.cis.McMaster.CA (Zdislav V. Kovarik)
Newsgroups: sci.math.num-analysis
Subject: Re: how parametrize d*d unitary matrices ?
Date: 13 Nov 1997 02:49:24 -0500

In article <64ckg6$2iso@moka.ccr.jussieu.fr>,
Michel PETITJEAN <ptitjean@ccr.jussieu.fr> wrote:
:How parametrize d*d unitary matrices when d>3 ? Is it known ?
:(quaternions are useful for 3*3 rotations, but do not apply for d>3)
:Thanks for references.
:Michel Petitjean,                     Email: petitjean@itodys.jussieu.fr
:ITODYS (CNRS, URA34)                         ptitjean@ccr.jussieu.fr

(Complex unitary matrices assumed) With the exception of a nowhere dense
set of zero measure, Cayley Transform helps:

Let U be unitary such that (-1) is not its eigenvalue. Then

 U = (I+H) * (I-H)^(-1)  where

 H = (U-I) * (U+I)^(-1)

is a skew-Hermitian matrix, and the set of skew-Hermitian matrices is a
real linear space, easy to parametrize. The dimension over reals is d^2 
(d for the diagonal, 2*d*(d-1)/2 for upper triangle - real parts and
imaginary parts).

My references are actually about the infinite-dimensional case, namely to
the treatment of spectral theory of Hermitian operators via reduction to
unitary operators:
 Krzysztof Maurin: Hilbert Space Methods (Polish Academy of Sciences)
and many other books on the subject.

Hope it helps, ZVK (Slavek).