Fréchet derivative of matrix-valued function

Question

If matrix-valued function $f : \mathbb{X} \times \mathbb{X} \to \mathbb{X}$ is defined by $(X, Y) \mapsto XY$, then how can we calculate the Fréchet derivative using the definition?

Note that $\mathbb{X} $ is the normed space of $ n\times n $ matrices with operator norm.

Answer 1

For any bilinear map $F$: $$DF(A,B): (X,Y)\longmapsto F(A,Y) + F(X,B).$$ In this case the Fréchet derivative is: $$(X,Y)\longmapsto AY + XB.$$ You can easily check this calculating $$(A+H)(B+K) - AB$$ for $H$, $K$ "small".