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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.2012-10-24
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    Additional details: A is a diagonal matrix and C is a symmetric matrix.2012-10-24

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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?2012-10-24
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    Well actually I am trying to find the second derivative of Tr(ABA'C) wrt A.2012-10-24
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    Do you know how a derivative is defined?2012-10-24
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    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 $D f(A) (\Delta) = \mathbb{tr} ( \Delta B A^TC + A B \Delta^T C)$.2012-10-24