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How to compute the derivative of $$\frac{\partial AB^T }{\partial{A}}$$ and $$\frac{\partial AB^T }{\partial{B}}$$

where $A \in R^{m \times n}$ and $B \in R^{r \times n}$.

Also, how can we analysis the dimension of final result? such as $\frac{\partial AB^T }{\partial{A}}$ is a matrix or tensor?

Thanks a lot.

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  • $\begingroup$ It will be 4th order tensor $\endgroup$ – user550103 May 4 '19 at 17:08
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Write the function using index notation (with the summation convention). $$\eqalign{ F_{ik} &= A_{ij} B_{jk}^T = A_{ij} B_{kj} \\ dF_{ik} &= dA_{ij}\,B_{kj} + A_{ij}\,dB_{kj} \\ }$$ Holding $B$ constant (i.e. $dB_{kj}=0$) yields the derivative with respect to $A$. $$\eqalign{ \frac{\partial F_{ik}}{\partial A_{pq}} &= (\delta_{ip}\delta_{jq})\,B_{kj} = \delta_{ip}\,B_{kq} \\ }$$ Similarly holding $A$ constant yields the derivative with respect to $B$. $$\eqalign{ \frac{\partial F_{ik}}{\partial B_{pq}} &= A_{ij}\,(\delta_{kp}\delta_{jq}) = A_{iq}\,\delta_{kp} \\ }$$ These derivatives require 4 indices for their description. So they are not matrices but 4th order tensors.

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