[SciPy-user] Re: finding approximate rank of matrix

Dan Christensen jdc at uwo.ca
Tue Aug 9 22:16:46 CDT 2005

Ok, my next question on this topic:

Suppose you have 300 vectors each with 64 components.  You have
determined using the singular value decomposition of the 300x64 matrix
they form that they all approximately lie in a 40-dimensional
subspace.  You suspect, however, that they really lie in the union of
two 30-dimensional subspaces (which intersect in a 20-dimensional
subspace).  How could you show this?

For example, suppose your vectors are in R^3 and all lie approximately
along either the x-axis or the y-axis.  The singular value
decomposition shows you that they all lie approximately in the
xy-plane.  Then, in this low-dimensional case, you can just notice
that every vector is a multiple of two chosen vectors.  But with
30-dimensional subspaces it's not clear how to choose spanning 
vectors for the hypothetical subspaces...

Thanks for any ideas,


PS: If it helps, I also can generate additional semi-random vectors
that lie in the same subspaces.  But I don't have a good way of
testing whether a particular vector is in those subspaces.

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