[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,
Dan
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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