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Some basic question about matrix calculus. Let $X$, $A$, $B$ be real matrices. Let $\operatorname{Tr}$ denote trace. Is \begin{equation} \frac{d }{dX} \operatorname{Tr}(X^T A XB) \end{equation} equal $(A+ A^T)XB$?

If not, How to compute it?

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    What exactly do you mean? Are you looking for the differenial $d_{X_0}\phi$ of the map $\phi:M_n(\mathbb R)\rightarrow \mathbb R,~X\mapsto \mathrm{Tr}(X^tAXB)$ at some point (i.e. matrix) $X_0$? If so, the answer will be a linear map from the matrices to the real numbers, not a matrix. You should proceed by steps. What is the differential of the matrix product inside the trace? What is the differential of the trace function? Then combine the two using the chain rule.2012-07-07
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    $\frac{d}{dX}$ is not notation that would universally make sense to mathematicians when $X$ is a matrix. That notation is only universally understood when $X$ takes numerical values. One reason for this is that $dX$ conceptually needs to be something "very small" that you could divide by; you cannot divide by matrices.2012-07-07
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    I would disagree. The notation is fairly standard.2012-07-07

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