Properties of the Matrix Exponential

In the settings of Single-Variable Calculus on an Interval and Euclidean Space and Real Matrices: Standing Notation, let $n\in\mathbb{N}$ and $M\in\mathcal{M}_{n}(\mathbb{R})$, and for $X\in\mathcal{M}_{n}(\mathbb{R})$ let $e^{X}$ be the exponential of $X$. Invertible matrices and the inverse are as in Inverse Matrix and Invertible Real Square Matrix (ii); differentiability of maps from an interval into $\mathcal{M}_{n}(\mathbb{R})$ and their derivatives are as in Derivative of a Vector-Valued or Matrix-Valued Map on an Interval (ii) and Derivative of a Vector-Valued or Matrix-Valued Map on an Interval (iii).

(i) $e^{0_{n}}=I_{n}$.

(ii) $(e^{M})^{\top}=e^{M^{\top}}$.

(iii) Let $B\in\mathcal{M}_{n}(\mathbb{R})$ with $BM=MB$. Then $Be^{M}=e^{M}B$.

(iv) Let $s,t\in\mathbb{R}$. Then $e^{(s+t)M}=e^{sM}e^{tM}$.

(v) $e^{M}$ is invertible, and $(e^{M})^{-1}=e^{-M}$.

(vi) Let $a,b\in\mathbb{R}$ with $a<b$. The map $[a,b]\to\mathcal{M}_{n}(\mathbb{R})$, $u\mapsto e^{uM}$, is differentiable, and its derivative at every $u\in[a,b]$ is $Me^{uM}=e^{uM}M$.

(vii) Let $A,B\in\mathcal{M}_{n}(\mathbb{R})$ with $AB=BA$. Then $e^{A+B}=e^{A}e^{B}$.