Matrix Derivative

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

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 $A \Delta C$.
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