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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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    I would disagree. The notation is fairly standard.2012-07-07

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The derivative of $F:X\mapsto X^T A X B =: F(X)$ is $D_V F (X) = V^TAXB+ X^TAVB$ while the derivative of $Y\mapsto Tr(Y)$ is simply

$D_V Tr(X)= Tr (V)$ Hence the derivative of your function is $D_V (Tr\circ F)(X) =DTr(F(X))D_VF(X) = Tr(V^TAXB+ X^TAVB)$

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    Using the various trace identities, you could also write $\mathbb{tr}(V^TAXB+ X^TAVB) = \mathbb{tr}((AXB+A^T X B^T)^T V)$, which shows that the 'gradient' is $AXB+A^T X B^T$.2012-07-07