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Can anyone please help me find the derivative of the ABC wrt B when:

A is say 3*3 matrix

B is 3*4 matrix

C is 4*4 matrix.

Thanks

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Could you please elaborate your question a little, perhaps even add a small example? Right now I don't really understand your question. Furthermore, it might be a good idea to show your own attempts at solving the problem. –  Ailurus Oct 24 '12 at 16:52
    
Additional details: A is a diagonal matrix and C is a symmetric matrix. –  star87fire Oct 24 '12 at 16:53
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1 Answer

up vote 0 down vote accepted

Let $\phi(B) = ABC$. $\phi$ is linear, so we have $\phi(B+\Delta) = \phi(B) + \phi(\Delta)$. It follows (Since $\phi(B+\Delta) - \phi(B) - \phi(\Delta) = 0$) that the derivative is $D\phi(B)(\Delta) = A \Delta C$.

This should be interpreted as the derivative of $\phi$ at the point $B$ in the direction $\Delta$ is $A \Delta C$.

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No, the algebraic formula is given above, there is no need to solve anything numerically. Any linear function (in finite dimensions) is its own derivative. Perhaps you could elaborate why you want the derivative? –  copper.hat Oct 24 '12 at 17:03
    
Well actually I am trying to find the second derivative of Tr(ABA'C) wrt A. –  star87fire Oct 24 '12 at 17:09
    
Do you know how a derivative is defined? –  Arkamis Oct 24 '12 at 17:09
    
If $f(A) = \mathbb{tr} (A B A^T C)$, then compute $F(A+\Delta)$, subtract $F(A)$ and find the linear terms (note that $\mathbb{tr}$ is linear). This will give $D f(A) (\Delta) = \mathbb{tr} ( \Delta B A^TC + A B \Delta^T C)$. –  copper.hat Oct 24 '12 at 17:39
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