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How does one approach finding $\nabla_X\|AXB\|_F^2$?

Normally I would expand to $\|AXB\|_F^2=\text{Trace}(B^TX^TA^TAXB)$ and rearrange using the cyclic property of traces to make $X$ the first and/or last matrix in the product, but this isn't possible here.

$A$ is wide and $B$ is tall; both are full-rank. $X$ is square. All matrices are real-valued.

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Look up formula (116) in the Matrix Cookbook:

$$\frac{\partial}{\partial X} \operatorname{tr}( B^T X^T C X B ) = C^T X B B^T + C X B B^T$$

So, the answer is:

$$2 A^T A X B B^T$$

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