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 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).