Derivative of Constant times norm of a matrix

Question

Compute: $$ \frac{\partial}{\partial W} \Vert c \times W^{T} \Vert $$ where c $\in$ $R^{1 \times n}$, i.e. a vector independent of W and W $\in$ $R^{n \times n}$, and the norm is a Frobenius norm.

Answer 1

Rather than a row vector, let $c$ be a column vector.
Find the differential, then the gradient of the function as $$\eqalign{ \phi^2 &= \|Wc\|^2 = Wc:Wc \cr 2\phi\,d\phi &= 2Wc:dW\,c \cr d\phi &= \frac{1}{\phi}Wcc^T:dW \cr \frac{\partial\phi}{\partial W} &= \frac{1}{\phi}Wcc^T= \frac{Wcc^T}{\|Wc\|} \cr }$$ In some of the steps above, a colon was used to denote the trace/Frobnius product, i.e. $$A:B = {\rm tr}(A^TB)$$ $$\eqalign{}$$ $$\eqalign{}$$ $$\eqalign{}$$