Directional derivative of matrix multiplication

Question

Suppose $U, A\in\mathbb{R}^{n\times n}$ and $f: \mathbb{R}^{n\times n}\rightarrow \mathbb{R}^{n\times n}$ is given by $f(X)=AX$. What is then directional derivative of $f$ in the direction $U$?

$f$ being linear, I suppose it is something like $AU$ or $UA$ (with perhaps taking a transpose or two), but I'm not sure what the exact expression should be.

Answer 1

As $f$ is linear, the total derivative of $f$ at any point is $f$ itself. The directional derivative in the direction $U$ would be the total derivative applied to $U$, i.e. the directional derivative is $f(U)$.

Answer 2

With the definition: the directional derivative of $f$ in $X$ in direction $U$ is

$= \lim_{t \to 0}\frac{f(X+tU)-f(X)}{t}=\lim_{t \to 0}\frac{AX+tAU-AX}{t}=AU.$