Trouble with little o notation.

Question

Problem: Compute the differential of $\phi(t) = e^{A+tB}$ at $0$ where $A$ and $B$ are commuting square matrices of size $n\times n.$

We proceed in the usual manner: $$\phi(0+h) = e^{A+hB} = e^{A}e^{hB}$$ since $A$ and $hB$ commute. Thus $$\phi(h) = e^{A}\left(I+hB+o(hB)\right)$$ $$ = e^A+h e^AB+e^{A}o(hB) = \phi(0)+he^{A}B+o(h).$$ I am not sure why $e^{A}o(hB)$ is $o(h).$ Perhaps someone can explain?

Answer 1

One definition of $o(h)$ is as follows:

A matrix valued function $f(h)$ is $o(h)$ if $\lim_{h \to 0} \frac{\|f(h)\|}{h} = 0$ (for some matrix norm $\|\cdot\|$).

With this definition, I think you should find the result easy to prove.