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Matrix calculus deals with the problems of differentiating (possibly matrix-valued) functions of matrices
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What is the derivative of $\|X^T-S^TAX^T\|_F^2$ w.r.t $A$?
What is the derivative of $F = \|X^T-S^TAX^T\|_F^2$ w.r.t $A$,
where $X \in\mathbb R^{d \times N}$, $S \in\mathbb R^{k \times N}$, and $A \in \mathbb{R}^{k \times N}$?
I have tried, and it is as fo …