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From: Chip Eastham on 22 Apr 2010 10:43
On Apr 22, 6:42 am, Cagdas Ozgenc <cagdas.ozg...(a)gmail.com> wrote:
> Greetings,
>
> I have the following problem at hand:
>
> A is an unknown mxn matrix where m<=n.
>
> {Xi} is a a set of nx1 vectors.
>
> I would like to determine matrix A from {Xi} where pairwise euclidean
> distances between A*Xi and A*Xj will be minimized for all i,j. I
> understand that there should be some constraint on components of A so
> that we don't end up with the trivial solution of a zero matrix.
>
> How do I solve this problem? How many Xs do I need? What should be the
> constraint on A? Othogonality?
>
> Thanks in advance

Perhaps the problem would make more sense if
you explain the motivation.

If A is mxn, it amounts to a linear mapping
from R^n to R^m. You ask how many Xs are
needed, so evidently you have some leeway
in what Xi's are to be chosen.

If the distance between A*Xi's is to be
minimized, then even without setting A = 0
one might have all the A*X's equal. If
the Xi's are linearly independent, then
nonzero A can be constructed that sends
all of them to A*Xi = Y, some common Y
in R^m that is not necessarily zero.

Since m < n is allowed, it isn't clear what
you are suggesting by "orthogonality" of A,
possibly that the rows of A have unit length
and are mutually perpendicular. In that case
the mapping would amount (as far as distances
go) to a projection on the Xi's. Again some
more information about the motivation of your
problem would probably clear up whether this
restriction on A is useful.

regards, chip
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