I am very confused reading all those symbols I do not understand, I am just trying to find partial derivative when two matrices are multiplied.
For an example, suppose that $A$ is a $3\times 4$ matrix and that $B$ is a $4\times 2$.
After I do multiplication $f(A,B) = AB$, I get a matrix $3\times 2$ matrix $C$.
Now my question is, if I try to find the partial derivative of $f(A,B)$ with respect to $B$, do I use the same rule as with scalars? I guess not, correct, so would anyone be so nice and tell to me a simple person how I could solve this. Please do not write in equations, because I can not read them! I understand mathematics only from intuition.
Each entry of the $3 \times 2$ matrix $C$ is a function of the entries of $B$, so we can take the partial derivative with respect to each entry of $B$.
Since $c_{ij} = a_{i1} b_{1j} + a_{i2} b_{2j} + a_{i3} b_{3j} + a_{i4} b_{4j}$, one could write $\frac{\partial c_{ij}}{\partial b_{kj}} = a_{ik}$ and $\frac{\partial c_{ij}}{\partial b_{k\ell}} = 0$ for $\ell \ne j$. Then you can somehow arrange this into a $3 \times 2 \times 4 \times 2$ array of partial derivatives and call this the "partial derivative of $f$ with respect to $B$."