Let $A$ and $B$ be two arbitrary matrices of $\mathcal{M}_d(\mathbf{R})$. (a) Let $D \in \mathcal{M}_d(\mathbf{R})$ such that $\|D\| \leq 1$. Show that there exists a constant $\mu > 0$ independent of $D$ such that $$\left\|\exp(D) - I_d - D\right\| \leq \mu \|D\|^2$$ (b) Show that there exists a constant $\nu > 0$, and for all $n \geq 1$ a matrix $C_n \in \mathcal{M}_d(\mathbf{R})$, such that $$\exp\left(\frac{A}{n}\right)\exp\left(\frac{B}{n}\right) = I_d + \frac{A}{n} + \frac{B}{n} + C_n \quad \text{and} \quad \|C_n\| \leq \frac{\nu}{n^2}$$
Let $A$ and $B$ be two arbitrary matrices of $\mathcal{M}_d(\mathbf{R})$.
(a) Let $D \in \mathcal{M}_d(\mathbf{R})$ such that $\|D\| \leq 1$. Show that there exists a constant $\mu > 0$ independent of $D$ such that
$$\left\|\exp(D) - I_d - D\right\| \leq \mu \|D\|^2$$
(b) Show that there exists a constant $\nu > 0$, and for all $n \geq 1$ a matrix $C_n \in \mathcal{M}_d(\mathbf{R})$, such that
$$\exp\left(\frac{A}{n}\right)\exp\left(\frac{B}{n}\right) = I_d + \frac{A}{n} + \frac{B}{n} + C_n \quad \text{and} \quad \|C_n\| \leq \frac{\nu}{n^2}$$