Eigensystem ordering

• To: mathgroup at yoda.physics.unc.edu (Mathematica User's Group)
• Subject: Eigensystem ordering
• From: Keith Clay <clay at galileo.phys.washington.edu>
• Date: Tue, 21 Sep 93 9:27:57 MDT

```The following question came to me from two fellow Mathematica users,
and I cannot find an answer in any of the standard literature:

> When you call Eigensystem[m], you get a result of the form:
>
> { n-vector, {vector1, vector2, ... , vector_n}}
>
> Q: Does the ith Eigenvalue in n-vector ALWAYS correspond to the ith
> Eigenvector in the list?  It usually does, but we want to know if this
> can always be assumed.  ONE OF US THINKS THIS SHOULD BE TRUE, ANOTHER
> THINKS THEY HAVE SEEN A CASE TO THE CONTRARY.

In other words, can we be sure this quantity is always zero

Eigensystem[m][[1,i]]*Eigensystem[m][[2,i]] - m.Eigensystem[m][[2,i]]

(with the same i used for the eigenvector and eigenvalue)?

Has anyone else seen a case to the contrary?

(Page 664 of The Book says that this is true for one matrix, for i=1.
Searching for exceptions has turned up none, but these people are looking
for certainty.)

It would certainly seem absurd to report the results any other way, but
I can't find any guarantees or details of how the output is put together.

Thanks.

------------------------------------------------------------------------

Keith Clay                                    Department of Physics, FM-15
(clay at galileo.phys.washington.edu)            University of Washington
( -or-  clay at phys.washington.edu )            Seattle, WA  98195

```

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