Re: order of eigenvalues

"Willem Geert " <wgphaff@xxxxxxxxx> wrote in message
<fnv8sh$puv$1@xxxxxxxxxxxxxxxxxx>...

Assume that you have two sets of eigenvectors
and eigenvalues, related by the fact that they
came from systems which were close to each
other. Now shuffle the eigenvalues and vectors
from system 2 so that its eigenvectors are
maximally aligned with those of system 1.

This seems like it might be easy, except that
the tiny eigenvalues for some systems may
sometimes end up with virtually random
eigenvectors. I'd visualize some potentially
nasty problems to resolve.


Thanks for the reply!

I've already looked into the eigenvector-approach. It works
most of the time. Since the eigenvectors returned by matlab
always have length 1, you can figure out which eigenvectors
are close together by inspecting the norm of the
difference. Basically, I use

normOfDifference = sum(abs(evectorOld - evectorNew));

as the measure of difference. I then figure out which
eigenvectors are the best match*. The assumption underlying
this however is that:

sum(abs(evector_i - evector_j)) >>> sum(abs(evector_i_now -
evector_i_previous)).

This fails every once in a while though.

Using the angle between the differenct is also an option,
but not based on the cosine; round-off error is too large.

Willem Geert

* Compensating for the fact that eigenvectors are only
defined up unto a scalar multiple, sometimes eig() also
returns them as the negative of the previous eigenvectors.

This is why the norm of the difference is not
the right method. instead, use the dot product
of the two vectors, and take the absolute value.

Two vectors which are close, even if the sign is
wrong on one of them, will still have an absolute
dot product near 1.

John
.

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