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From: paolo serena on 2 Nov 2009 03:08
Hi to everybody.

I have the same problem of finding the optimal rotation matrix, but in a complex space. Hence the rotation matrix is actually a unitary matrix of size 2x2 of complex entries. Can I use the same procedure of Mr Stafford with the transpose-conjugate operator instead of only transpose?
Concerning the problem of determinant +/-1 of this post (see bottom), I found a very nice solution at:

http://www.kwon3d.com/theory/jkinem/rotmat.html

In my problem with complex data the determinant is exp(i*phi), hence I don't know how the previous problem of the determinant maps in my case. Is still a problem, or the proposed solution Q = V*U' is always fine in my case?

Thanks
Paolo

> If X and Y are the two n x m data sets in question where n is the number of
> data items and m the number of components to the data, then one step of
> the procrustes procedure is to do a singular value decomposition, 'svd', on
> the m x m matrix X'*Y, getting
>
> X'*Y = U*S*V'
>
> where U and V are unitary matrices and S is diagonal with the non-negative
> singular values. The best fitting unitary transformation in the least squares
> sense is given by Q = V*U'. However, if it turns out that Q involves a
> reflection, that is, its determinant is -1, then what you need to do is reverse
> the sign in the rightmost column of either U or V, corresponding to the
> smallest singular value.
From: paolo serena on 2 Nov 2009 03:09
Hi to everybody.

I have the same problem of finding the optimal rotation matrix, but in a complex space. Hence the rotation matrix is actually a unitary matrix of size 2x2 of complex entries. Can I use the same procedure of Mr Stafford with the transpose-conjugate operator instead of only transpose?
Concerning the problem of determinant +/-1 of this post (see bottom), I found a very nice solution at:

http://www.kwon3d.com/theory/jkinem/rotmat.html

In my problem with complex data the determinant is exp(i*phi), hence I don't know how the previous problem of the determinant maps in my case. Is still a problem, or the proposed solution Q = V*U' is always fine in my case?

Thanks
Paolo

> If X and Y are the two n x m data sets in question where n is the number of
> data items and m the number of components to the data, then one step of
> the procrustes procedure is to do a singular value decomposition, 'svd', on
> the m x m matrix X'*Y, getting
>
> X'*Y = U*S*V'
>
> where U and V are unitary matrices and S is diagonal with the non-negative
> singular values. The best fitting unitary transformation in the least squares
> sense is given by Q = V*U'. However, if it turns out that Q involves a
> reflection, that is, its determinant is -1, then what you need to do is reverse
> the sign in the rightmost column of either U or V, corresponding to the
> smallest singular value.
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