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Given $n\times n$ matrix $A$, find (with proof) the gradient $\nabla_X \mbox{tr} (XAX)$. The matrix $A$ does not depend on $X$.

I know that the final answer is of the form:

$$\frac{\partial \mbox{tr} }{\partial X_{ij}}= \sum_k A_{jk}X_{ki}+\sum_k X_{jk} A_{ki}=(AX+XA)_{ji}$$

Is there a way to write out the partial derivative of the individual entries of the matrices?

Rodrigo de Azevedo
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Mike
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    Welcome to MSE. Your question is phrased as an isolated problem, without any further information or context. This does not match many users' quality standards, so it may attract downvotes, or be closed. To prevent that, please [edit] the question. This will help you recognize and resolve the issues. Concretely: please provide context, and include your work and thoughts on the problem. These changes can help in formulating more appropriate answers. – José Carlos Santos Oct 28 '22 at 11:10
  • @JoséCarlosSantos The statement "I know the form of the final answer but I'm looking for a way to write out the partial derivatives" seems like sufficient context to me – Ben Grossmann Oct 28 '22 at 14:31
  • @Mike you've already written that $\frac{\partial f}{\partial X_{ij}} = \sum_k A_{jk}X_{ki}+\sum_k X_{jk} A_{ki}$. It looks a lot to me like you have already "written out the partial derivative of the individual entries of the matrices". If you haven't already done that, then it's not clear to me what you mean by that phrase. – Ben Grossmann Oct 28 '22 at 14:46
  • Related – Rodrigo de Azevedo Oct 29 '22 at 16:22

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