Re: complex orthogonal group

[A complimentary Cc of this posting was sent to
Tobias Fritz
<tfritz@xxxxxxxxxxxxxxx>], who wrote in article <Pine.LNX.4.62.0510011730420.19741@xxxxxxxxxxxxxxxxxxx>:
> Is SO(n,C) generated by reflections (at non-null vectors)? Any hints or
> references would be appreciated!

> Since there is no spectral theorem due to lack of definiteness, this seems
> a lot harder than the real case, so I have no idea.

Any good thorough book on linear algebra (i.e., designed not for
teaching students) will contain the spectral theorem over an
algebraically closed case. There is also a classification in the real
case. One of the most comprehensive compendium of solutions to
similar problems is the exposition

@article {Thom91Pen,
old = MR91k:15031,
AUTHOR = {Thompson, Robert C.},
TITLE = {Pencils of complex and real symmetric and skew matrices},
JOURNAL = {Linear Algebra Appl.},
FJOURNAL = {Linear Algebra and its Applications},
VOLUME = {147},
YEAR = {1991},
PAGES = {323--371},
ISSN = {0024-3795},
CODEN = {LAAPAW},
}

Hope this helps,
Ilya

P.S. I expect that your question would allow a simpler solution than
going through the classification - although the classification
is not harder than Jordan one.

.