Matrix Exponential

We define the Matrix Exponential, denoted by $e^X$. For a generic matrix pair $X,Y\in M_n(\mathbb{C})$, the matrix exponential is defined by:

1. $e^0=1$

2. $e^{\overline{X}}=\overline{(e^X)}$ and $e^{X^{\ast }}=(e^X{)}^{\ast }$, where $\overline{X}$ is the Transpose and $X^{\ast }$ is the complex conjugate.

3. $e^{AX{A}^{-1}}=A{e}^X{A}^{-1}$ holds $\forall A\in G{L}_n(\mathbb{C})$.

4. $\det (e^X)=e^{\mathrm{Tr}[X]}$, where $\mathrm{Tr}[\cdot ]$ is the Trace

5. $XY=YX\Rightarrow e^{X+Y}=e^X{e}^Y$

6. $e^X\in G{L}_n(\mathbb{C})\forall X$, i.e. $e^X$ is always an invertible matrix

7. $(e^X{)}^{-1}=e^{-X}$

8. $e^{X+Y}=\underset{m\to \infty }{\lim }{\left(e^{X/m}{e}^{Y/m}\right)}^m\forall X,Y\in M_n\mathbb{C}$

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