Re: complex orthogonal group

[A complimentary Cc of this posting was sent to
Tobias Fritz
<tfritz@xxxxxxxxxxxxxxx>], who wrote in article <Pine.LNX.4.62.0510032230470.13099@xxxxxxxxxxxxxxxxxxx>:

You send a "blind Cc" copy (as not marked as a copy of a posting).
Now I need to copy my reply to you here. Please do not send blind Cc's.

> >Any good thorough book on linear algebra (i.e., designed not for
> >teaching students) will contain the spectral theorem over an
> >algebraically closed case.

> Though I have to admit that I did not check really thoroughly, I
> doubt that this is true. A key assumption in the proof of the
> standard (finite-dim) spectral theorem for normal operators is that
> the orthogonal complement of a subspace has trivial intersection
> with that subspace; this fails for the C^n case, since it has null
> vectors.

The conclusion of the complex spectral theorem is different (comparing
to the positive real case). As I said, in addition to 1x1 Jordan
block (as in real positive case) larger blocks can appear.

But each block has (in addition to its size) exactly one parameter:
the eigenvalue - exactly as in the real positive case.

Hope this helps,
Ilya

.

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