\[\begin{align}
  & {{e}^{At}}=\left( \begin{matrix}
   -{{e}^{t}}+{{e}^{2t}} & -{{e}^{t}}+{{e}^{3t}}  \\
   2{{e}^{t}}+{{e}^{2t}} & -{{e}^{2t}}+{{e}^{3t}}  \\
\end{matrix} \right) \\
& A{{e}^{At}}=\left( \begin{matrix}
   -{{e}^{t}}+2{{e}^{2t}} & -{{e}^{t}}+3{{e}^{3t}}  \\
   2{{e}^{t}}+2{{e}^{2t}} & -2{{e}^{2t}}+3{{e}^{3t}}  \\
\end{matrix} \right) \\
&  \\
& t=0 \\
& \text{               }A=\left( \begin{matrix}
   1 & 2  \\
   4 & 1  \\
\end{matrix} \right) \\
&  \\
& {{e}^{A}} \\
& t=1 \\
& {{e}^{A}}=\left( \begin{matrix}
   -e+{{e}^{2}} & -e+{{e}^{3}}  \\
   2e+{{e}^{2}} & -{{e}^{2}}+{{e}^{3}}  \\
\end{matrix} \right)\doteq \left( \begin{matrix}
   4.67 & 17.37  \\
   12.83 & 12.70  \\
\end{matrix} \right) \\
& A=\left( \begin{matrix}
   -\frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}}  \\
   1 & 1  \\
\end{matrix} \right)\left( \begin{matrix}
   1-2\sqrt{2} & 0  \\
   0 & 1+2\sqrt{2}  \\
\end{matrix} \right){{\left( \begin{matrix}
   -\frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}}  \\
   1 & 1  \\
\end{matrix} \right)}^{-1}} \\
& {{e}^{A}}=\left( \begin{matrix}
   \frac{1}{2}{{e}^{1-2\sqrt{2}}}\left( 1+{{e}^{4\sqrt{2}}} \right) & \frac{{{e}^{1-2\sqrt{2}}}\left( -1+{{e}^{4\sqrt{2}}} \right)}{2\sqrt{2}}  \\
   \frac{{{e}^{1-2\sqrt{2}}}\left( -1+{{e}^{4\sqrt{2}}} \right)}{\sqrt{2}} & \frac{1}{2}{{e}^{1-2\sqrt{2}}}\left( 1+{{e}^{4\sqrt{2}}} \right)  \\
\end{matrix} \right)\doteq \left( \begin{matrix}
   23.07 & 16.20  \\
   32.4 & 23.07  \\
\end{matrix} \right) \\
&  \\
& {{e}^{A}} \\
\end{align}\]

2Â¥: Originally posted by zaq123321 at 2015-01-13 00:59:37
The input seems not correct since it violates exp(0)=I.

3Â¥: Originally posted by kabaisun at 2015-01-13 09:31:32
Äĸöinput?
...

4Â¥: Originally posted by pippi6 at 2015-01-13 12:52:02
2Â¥ÊÇ˵ÔÚt=0ʱ exp(At) =exp(0)=I ӦΪµ¥Î»¾ØÕ󣬿ÉÊÇÖ±½ÓÑéÖ¤Äã±í´ïʽ
exp(A 0) /= I. ÊDz»ÊÇ a21 ÀïÃæexp(2t)µÄ·ûºÅ´íÁË£¿...