[Eeglablist] Eigenvectors, reflections and SVD

[Luca Finelli]

Dear Marco

	On Wed, 8 Dec 2004, Marco Congedo wrote:

	> Hello,
	> 
	>  I have a question on Eigenvectors, reflection and SVD.
	> 
	> Given a square non-symmetric matrix A, consider the Singular Value
	> Decomposition (SVD) of its symmetric part B=1/2(A+A'), as
	> 
	>  SVD(B)+UwV'
	> 
	> U and V, both orthogonal, should now be equal, hence projecting
	> in the same space. However I noticed that V
	> have some rows with sign reversed as compared to U.

Algebraically, for a real square symmetric matrix A you can find a 
decomposition 

	A = R S R'

R orthogonal, the columns forming an orthonormal basis of eigenvectors of 
A. 
However, in general, SVD is unique for a given arbitrary matrix up to a 
sign change in column pairs u(i) and v(i).

	> It seems that the spaces for the U' projection and V' projection
	> are identical, out of a reflection of one or more axes.

Right. And a space does not change if one or more basis vectors are 
flipped (as long as the coordinates of vectors in that space are changed 
accordingly).

	> My question is, is there some information in this "reflection" and if so
	> what it means, or it is just an artifact of the SVD algorithm?

>From what above, we may conclude that it is an "artifact" of the 
SVD algorithm, which is designed for arbitrary rather than square 
symmetric matrices.

A geometrical interpretation follows if we describe the SVD decomposition 
stepwise as a rotation, scaling and a second rotation. 
In your particular case the first rotation is the inverse of the second (R 
orthogonal:  R' = inv(R)).
The first rotation rotates the n-dimensional space as to carry the 
orthonormal basis of eigenvectors of A into the standard basis. S 
scales the basis vectors accordingly. Finally the space is 
backrotated so that the scaled basis vectors are rotated into the images 
under A of the orthonormal basis of eigenvectors of A.

Now, flipping signs in some columns of R flips the basis vectors, 
therefore changes the rotation, and the backrotation, accordingly.

For imaging applications, the undetermined sign of eigenvectors 
corresponds to the undetermined polarity of eigenimages (columns of V) and 
their corresponding timing weights (columns of U).

Hope this helps.
Luca


	> 
	> Thank you very much.
	> 
	> Marco Congedo
 
________________________________________________
 Luca A. Finelli, Ph.D.
                                                                                                          
 Computational Neurobiology Laboratory
 The Salk Institute
                                                                                                          
 Swartz Center for Computational Neuroscience
 UCSD
                                                                                                          
 La Jolla CA 92037 - USA
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