Haven

2020-12-28

Use the definition of the matrix exponential to compute eA for each of the following matrices:
$A=\left[\begin{array}{ccc}1& 0& -1\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$

Theodore Schwartz

Step 1
Assumed matrix is
$A=\left[\begin{array}{ccc}1& 0& -1\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$
Step 2 Now ${A}^{2}=A\cdot A$
$=\left[\begin{array}{ccc}1& 0& -1\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]\cdot \left[\begin{array}{ccc}1& 0& -1\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$
$=\left[\begin{array}{ccc}1& 0& -2\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$
${A}^{3}={A}^{2}\cdot A$

$=\left[\begin{array}{ccc}1& 0& -3\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$
$\dots$
$\text{In general}$

Step 3
The matrix exponential's definition now shows that
${e}^{A}=I+A+\frac{{A}^{2}}{2!}+\frac{{A}^{3}}{3!}+\dots$
$=\left[\begin{array}{ccc}1& 0& 1\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]+\left[\begin{array}{ccc}1& 0& -1\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]+\frac{1}{2!}\left[\begin{array}{ccc}1& 0& -2\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]+\frac{1}{3!}\left[\begin{array}{ccc}1& 0& -3\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]+\dots$
$=\left[\begin{array}{ccc}1+1+\frac{1}{2!}+\frac{1}{3!}+\dots & 0& 0-\left(1+1+\frac{1}{2!}+\frac{1}{2!}+\dots \right)\\ 0& 1+1+\frac{1}{2!}+\frac{1}{2!}+\dots & 0\\ 0& 0& 1+1+\frac{1}{2!}+\frac{1}{2!}+\dots \end{array}\right]$
$=\left[\begin{array}{ccc}e& 0& -e\\ 0& e& 0\\ 0& 0& e\end{array}\right]$
Step 4
${e}^{A}=\left[\begin{array}{ccc}e& 0& -e\\ 0& e& 0\\ 0& 0& e\end{array}\right]$

Jeffrey Jordon