Derivative of a matrix times vector with respect to an entry of the matrix

Question

I'm trying to find the derivative of $$\frac{\partial{A\vec{x}}}{\partial{A_{ij}}}$$ but I'm having some trouble figuring it out. $A\vec{x}$ will be a vector where the $i^{th}$ entry is $$A_{i1}x_1 + A_{i2}x_2 ...$$ Will the resulting derivative be a vector with 0's except in the $i^{th}$ entry where there is $x_j$? Thank you!

Answer 1

The matrix $\frac{\partial A}{\partial A_{ij}}$ is a matrix with vanishing entries, except the entry $(i,j)$ which is $1$. Therefore, its product with $\vec{x}$ is the vector $(0,...,0,x_j, 0,...,0)$ where $x_j$ is in position $i$.