derivative a portion of a matrix

Question

I have a matrix $Q \in R^{m,n}$ and its derivative w.r.t a parameter $Q'\in R^{m,n}$. Let say that at some point, I needed to use a portion of the matrix $Q$ (example: the first column of $Q$), and I want to know the derivative of this vector column, is the first column of $Q'$ is the derivative of the first column of $Q$.

Answer 1

Yes. Consider the $k^{th}$ column of $Q$ and its derivative $$\eqalign{ q_k &= Q\,e_k \cr dq_k &= dQ\,e_k + Q\,de_k \cr }$$ Since the standard basis vectors $\{e_k\}$ are constant, we have $$\eqalign{ dq_k &= dQ\,e_k \cr }$$ as you anticipated.