In article <[EMAIL PROTECTED]>,
Fernando De la Torre <[EMAIL PROTECTED]> wrote:
>Hi all,
>I want to make the matrix derivative of an expression like:
>min over B of sum_i A_i B(B^TW_iB^T)^-1 B^TA_i subject to
>B^TB=Identity.
>where B^T denotes transpose of B and it is an arbitrary matrix.
>Do you know any good reference book which show how to make derivatives
>w.r.t. the pseudo-inverse or in general
>any good reference for matrix derivatives. Or any goo trick to minimize a
>similar expression w.r.t. matrix B with constraints?
>Thanks very much in advance. If possible send an e-mail to [EMAIL PROTECTED]
>Take care.
Derivatives do not work, but differentials do. It will be a
little easier to read if ' is used for transpose.
The Frechet differential of phi(B) is given by
(dphi)(B, dB) = lim_{c->0} (phi(B+cdB) - phi(B))/c
Differentials satisfy the usual laws, but one cannot ignore
the order of multiplication of matrices. So the condition
B'B = I yields dB'*B + B'*dB = 0, or B'*dB is skew symmetric.
Likewise, A*A^-1 = I yields d(A^-1) = - A^-1*dA*A^-1.
I could write out the full differential. As for minimizing,
the differential must be 0 for all possible values of dB
subject to the conditions, so you could use the condition
to write dB = B*S, where S is skew, and then the resulting
expression would have to be 0 for all 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, Deptartment of Statistics, Purdue University
[EMAIL PROTECTED] Phone: (765)494-6054 FAX: (765)494-0558
.
.
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