Re: Eigensystem applied to a unitary matrix crashes Mathematica 4.

• To: mathgroup at smc.vnet.net
• Subject: [mg21335] Re: Eigensystem applied to a unitary matrix crashes Mathematica 4.
• From: sidles at u.washington.edu (John A. Sidles)
• Date: Sun, 26 Dec 1999 01:26:13 -0500 (EST)
• Organization: University of Washington, Seattle
• References: <831tqg\$fmh@smc.vnet.net> <83cn2s\$5o2\$8@dragonfly.wolfram.com> <83ncc5\$5ph\$4@dragonfly.wolfram.com>
• Sender: owner-wri-mathgroup at wolfram.com

```In article <83ncc5\$5ph\$4 at dragonfly.wolfram.com>,
Kevin J. McCann <kevinmccann at Home.com> wrote:
>Well it looks like Ingolf has got himself a winner here.  I tried it on my
>W98 system and Commands 1,2,3 crash the kernel.  Further, command 1 followed
>by repeated execution of the Eigensystem command gives different results
>each time.
>
>Kevin

Just to be fair, you gotta read the Eigensystem[] documentation
with the care of a tax lawyer!  As Mathematica support staff
will confirm, the following behaviors are all legal for
Eigensystem[], are *not* regarded as bugs by Wolfram Research,
and must be respected by anyone who wants to write reliable code:

Eigensystem[(a general Hermitian matrix)] =>
eigenvectors which are
(1) complex
(2) non-orthonormal
(3) different every time Eigensystem[] is called
(4) sensitive to least-significant bits in the input.

Eigensystem[] *does* guarantee that the returned eigenvectors
will be complete; this is consistent with the above behaviors.

As is well known, it a well-posed numerical problem to
calculate the real orthonormal eigenvectors of a general
Hermitian matrix, and Eigensystem[] *usually* does so, but
does sporadically return eigenvectors with the above -- more
general -- characteristics; this is just a "feature" of life
in the Mathematica world.

I have no complaints about the above, but I do think the
Eigensystem[] documentation might reasonably be revised to
be a little more forthcoming about these behaviors, which
over the years have taken so many users by surprise.

```

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