[gmx-users] subspaces / cov. matrices overlap

Bert de Groot bgroot at gwdg.de
Wed Apr 3 10:23:55 CEST 2002

Jose D Faraldo-Gomez wrote:

> Thanks Bert, I see your point. However, I don't understand how/why the
> complete sets of eigenvectors in $trr and $trr2 necessarily span the same
> subspace... even though in my case they correspond to (a) ligand-free and a
> ligand-bound forms of a protein, and (b) different sampling windows - e.g.
> 100ps vs 10ns. I guess that's why I was wondering whether this is related to
> any mathematical property of the eigenvectors...

If it is the same system (same number of degrees of freedom, ie protein-only
in your case for example), then the total configurational space spanned by 
these coordinates is identical in both cases, and the sampling in these spaces
can be directly compared to each other, after removal of overall translation and 
rotation (ie after fitting to a reference structure). Remember that it's the 
coordinates that span the space (3N dimensional for N atoms), and that is therefore
defined even without having a trajectory or ensemble. The trajectory or
ensemble of structrures represent a cloud WITHIN this space. This is by definition, 
and is not a special property of the eigenvectors. The eigenvectors are just a 
transformation of the axes that span this space in such a way that most of the
sampled region is concentrated along eigenvectors with large eigenvalues. The
way the eigenvectors are defined and ordered therefore is dependent on the
trajectory or ensemble over which you calculated them, but taken all togehter,
they will still span the same complete configurational space. The eigenvectors
are therefore nothing else than a convenient transformation, such that we
can concentrate on a small number of degrees of freedom, the collection of 
large-eigenvalue eigenvectors, which we also call the essential subspace, in which
most of the fluctuation takes place. The interesting comparison therefore is to
see how similar these essential subspaces are for different simulations. Comparison
of individual eigenvectors usually doesn't make that much sense because in most
practical cases their definition will not be converged yet. We found in many cases,
however, that the essential subspace converges much faster than the individual
vectors that span it.


Dr. Bert de Groot

Max Planck Institute for Biophysical Chemistry
Theoretical molecular biophysics group
Am Fassberg 11 
37077 Goettingen, Germany

tel: +49-551-2011306, fax: +49-551-2011089

email: bgroot at gwdg.de

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