Hello I am working on the Matrix Computation Book by Golub and Van Loan and am stuck on a question.
Suppose $A \in \mathbb R^{n\times n} $ is nonsingular, b $\in \mathbb R^n, Ax = b $ and $C=A^{-1} $. Using the Sherman-Morrison formula to show that
$$\frac{∂x_k}{∂a_{ij}} = -x_j c_{ki}$$
So by using the Sherman-Morrison I understand I can get $x=$ something however from the Sherman-Morrison, how would I choose u and v to get that results?
For reference the Sherman-Morrison I am using is $$(A + uv^T)^{-1} = A^{-1} - \frac{A^{-1}uv^TA^{-1}}{1+v^TA^{-1}u}$$ thus for the question I would have $$(A + uv^T)^{-1} = C - \frac{Cuv^TC}{1+v^TCu}$$
Thanks for your time.
One method to compute $\frac{\partial f}{\partial a_{ij}}$ is to note that $$ f(A + h\,e_ie_j^T) = f(A) + h\frac{\partial f}{\partial a_{ij}} + O(h^2) $$ since $e_ie_j^T$ is the matrix with a $1$ in the $i,j$ slot and $0$s everywhere else.