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From: Vachel on 25 May 2010 09:32
On May 25, 1:49 am, Herman Rubin <hru...(a)skew.stat.purdue.edu> wrote:
> On 2010-05-23, Vachel <liangju...(a)gmail.com> wrote:
>
> > suppos matrix A = X-GSF' , and J = (||A||F ) 2 = tr(AA*) is the square
> > of A's Frobenius norm. now how to calculate the derivative of J
> > respect to S?
> > because J = tr(AA*), and d(J)/d(A) = 2A , can i calculate it like the
> > following?
> > d(J)/d(S) = [d(J)/d(A)] [d(A)/d(S)] ? it seems not right.
> > how to solve this problem?
> > is there any books or articles on this subject?
> > any suggestions can help!
> > thanks for anything useful!
>
> In general it is necessary to use differentials.  The
> differential of J is tr(dA*A')+tr(A*dA')=2tr(A'*dA).
> Now dA=-G*dS*F', so we get
>
>         dJ=-2tr(A'*G*dS*F') = -2tr(F'*A'*G*dS),
>
> which shows how to get the derivative of J with
> respect to any element of S.
>
> --
> This address is for information only.  I do not claim that these views
> are those of the Statistics Department or of Purdue University.
> Herman Rubin, Department of Statistics, Purdue University
> hru...(a)stat.purdue.edu         Phone: (765)494-6054   FAX: (765)494-0558

thanks !
but I've got another problem about the differentials
is d( tr(A) ) = tr( dA ) right? because i think the left is a
matrix ,while the right is a scalar.
so why  the differential of J is tr(dA*A')+tr(A*dA')=2tr(A'*dA) ?
and as i know, the dA = -F⊗G dS which is the Kronecker product.
is -F⊗G dS equal -G*dS*F ?


From: Vachel on 25 May 2010 09:35
is it necessary to use the vec() operator?
From: Vachel on 25 May 2010 12:35
On May 24, 3:46 am, Vachel <liangju...(a)gmail.com> wrote:
> suppos matrix A = X-GSF' , and J = (||A||F ) 2 = tr(AA*) is the square
> of A's Frobenius norm. now how to calculate the derivative of J
> respect to S?
>
> because J = tr(AA*), and d(J)/d(A) = 2A , can i calculate it like the
> following?
> d(J)/d(S) = [d(J)/d(A)] [d(A)/d(S)] ? it seems not right.
> how to solve this problem?
> is there any books or articles on this subject?
> any suggestions can help!
> thanks for anything useful!

is the following the final result?
dJ/dS = 2(X-GSF')(-F⊗G)
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