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Re: How to calculate the derivative of matrix w.r.t another matrix?

In article <c215ts$8gh$1 at>, "Fred" <f0z6305 at> 

> I have a problem of calculating the derivative of dxm matrix A with respect
> to another dxm matrix B,
> where A= [a1 a2 ... am] and B =[b1 b2 ... bm] with
> ai and bj are vectors.

one matrix with respect to another! That would mean something else 
altogether. It looks like you are after an element by element operation.

> Actually the matrix A itself is the first order derivative
> of a scalar J with respect to B, i.e., A = dJ/dB,
> where a1 = dJ/db1, a2 = dJ/db1, and so on.

This is an extension of the gradient operator. For example, for 
specified b, say

   b = {{x, y}, {z, t}}; 

and j, say

  j = x^2 + y^2 + z^2 - c t^2; 

you can compute dj/db using Outer: 

   a = First[Outer[D, {j}, b]]

> Now dA/dB is the second order derivative
> dA/dB = dJ^2/(dBdB')
>            = [dJ/db1 dJ/db1 ... dJ/dbm]/d[b1 b2 ... bm].
> So anybody have some idea on how to derive the formulation of dA/dB?

Again, this is just an Outer product:

   Outer[D, a, b]


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