matrix differentiation

Question

What is the derivative of $\mathbf{A}\mathbf{B}\mathbf{c}$ respect to $\mathbf{B}$, where $\mathbf{A}$ is a $n\times m$ matrix, $\mathbf{B}$ is a $m\times k$ matrix, and $\mathbf{c}$ is a $k\times 1$ vector.

Answer 1

Note that the map $\def\M#1#2{\operatorname{Mat}_{#1,#2}(\mathbf R)}$ $$ f: \M mk \to \def\R{\mathbf R}\R^n, \qquad B \mapsto ABc $$ is linear, hence the derivative is constant, it is given at any $B \in \M mk$ by $f$ itself, that is $$ f'(B): \M nk \to \R^n, \qquad H \mapsto AHc $$

Answer 2

Assume a single value decomposition ${\bf B} ={\bf U_{m\times m}\Sigma_{m\times k} V^*_{k\times k}}$.

Define a function $f({\bf B})=\bf{ABc}$ then, that is $$f'({\bf B})=\lim_{h\to0}\frac{{\bf ABc} - {\bf A U}({\bf\Sigma}+ h {\bf I_{m\times k}}) \bf{V^*}{\bf c}}{h}= \lim_{h\to0}\frac{{\bf ABc} - {\bf ABc}+ h{\bf A U I_{m\times k}V^* c}}{h}= {\bf A U I_{m\times k}V^* c} $$

This is almost the definition of the derivative, just instead of perturbing $(+h)$ the matrix its singular values being perturbed.