(1)

\begin{align*} e^{O} & =\sum_{k=0}^{\infty}\frac{O^{k}}{k!}\\ & =\sum_{k=0}^{\infty}\frac{I\delta_{0,k}}{k!}\\ & =I \end{align*}

(2)

\begin{align*} e^{aA}e^{bA} & =\sum_{k=0}^{\infty}\frac{\left(aA\right)^{k}}{k!}\sum_{j=0}^{\infty}\frac{\left(bA\right)^{j}}{j!}\\ & =\sum_{k=0}^{\infty}\sum_{j=0}^{\infty}\frac{a^{k}b^{j}A^{k+j}}{k!j!}\\ & =\sum_{k=0}^{\infty}\sum_{j=0}^{k}\frac{a^{k-j}b^{j}A^{k}}{\left(k-j\right)!j!}\cmt{k+j\rightarrow k,j\rightarrow j}\\ & =\sum_{k=0}^{\infty}\sum_{j=0}^{k}C\left(k,j\right)a^{k-j}b^{j}\frac{A^{k}}{k!}\\ & =\sum_{k=0}^{\infty}\left(a+b\right)^{k}\frac{A^{k}}{k!}\\ & =\sum_{k=0}^{\infty}\frac{\left(\left(a+b\right)A\right)^{k}}{k!}\\ & =e^{\left(a+b\right)A} \end{align*}

(3)

\begin{align*} e^{A}e^{-A} & =e^{A-A}\cmt{\because\left[A,A\right]=0}\\ & =e^{O}\\ & =I \end{align*} となるので
\[ e^{-A}=\left(e^{A}\right)^{-1} \] となる。

(4)

\begin{align*} e^{A}e^{B} & =\sum_{k=0}^{\infty}\frac{A^{k}}{k!}\sum_{j=0}^{\infty}\frac{B^{j}}{j!}\\ & =\sum_{j=0}^{\infty}\frac{B^{j}}{j!}\sum_{k=0}^{\infty}\frac{A^{k}}{k!}\\ & =e^{B}e^{A} \end{align*} \begin{align*} e^{A}e^{B} & =\sum_{k=0}^{\infty}\frac{A^{k}}{k!}\sum_{j=0}^{\infty}\frac{B^{j}}{j!}\\ & =\sum_{k=0}^{\infty}\sum_{j=0}^{\infty}\frac{A^{k}B^{j}}{k!j!}\\ & =\sum_{k=0}^{\infty}\sum_{j=0}^{k}\frac{A^{k-j}B^{j}}{\left(k-j\right)!j!}\cmt{k+j\rightarrow k,j\rightarrow j}\\ & =\sum_{k=0}^{\infty}\frac{1}{k!}\sum_{j=0}^{k}C\left(k,j\right)A^{k-j}B^{j}\\ & =\sum_{k=0}^{\infty}\frac{1}{k!}\left(A+B\right)^{k}\\ & =e^{A+B} \end{align*}

(5)

\begin{align*} e^{P^{-1}AP} & =\sum_{k=0}^{\infty}\frac{\left(P^{-1}AP\right)^{k}}{k!}\\ & =P^{-1}\left(\sum_{k=0}^{\infty}\frac{A}{k!}\right)P\\ & =P^{-1}e^{A}P \end{align*}

(6)

\begin{align*} \exp\left(A^{T}\right) & =\sum_{k=0}^{\infty}\frac{\left(A^{T}\right)^{k}}{k!}\\ & =\sum_{k=0}^{\infty}\frac{\left(A^{k}\right)^{T}}{k!}\\ & =\left(\sum_{k=0}^{\infty}\frac{A^{k}}{k!}\right)^{T}\\ & =\exp^{T}\left(A\right) \end{align*}

(7)

\begin{align*} \exp\left(\overline{A}\right) & =\sum_{k=0}^{\infty}\frac{\left(\overline{A}\right)^{k}}{k!}\\ & =\sum_{k=0}^{\infty}\frac{\overline{A^{k}}}{k!}\\ & =\overline{\sum_{k=0}^{\infty}\frac{A^{k}}{k!}}\\ & =\overline{\exp\left(A\right)} \end{align*}

(8)

\begin{align*} \exp\left(A^{*}\right) & =\exp\left(\overline{A^{T}}\right)\\ & =\overline{\exp\left(A^{T}\right)}\\ & =\overline{\exp^{T}\left(A\right)}\\ & =\exp^{*}\left(A\right) \end{align*}