Consider a linear system whose state space representation is $\dot{\mathbf{x}}(t)=\mathbf{A} \mathbf{x}(t)$. If the initial state vector of the system is $\mathbf{x}(0)=\left[\begin{array}{r}1 \\ -2\end{array}\right]$, then the system response is $\mathbf{x}(t)=\left[\begin{array}{r}e^{-2 t} \\ -2 e^{-2 t}\end{array}\right]$
If the initial state vector of the system changes to $\mathbf{x}(0)=\left[\begin{array}{r}1 \\ -1\end{array}\right]$, then the system response becomes $\mathbf{x}(t)=\left[\begin{array}{c}e^{-t} \\ -e^{-t}\end{array}\right]$.
The eigenvalue and eigenvector pairs $\left(\lambda_{i}, v_{i}\right)$ for the system are
- $\left(-1,\left[\begin{array}{r}1 \\ -1\end{array}\right]\right)$ and $\left(-2,\left[\begin{array}{r}1 \\ -2\end{array}\right]\right)$
- $\left(-2,\left[\begin{array}{r}1 \\ -1\end{array}\right]\right)$ and $\left(-1,\left[\begin{array}{r}1 \\ -2\end{array}\right]\right)$
- $\left(-1,\left[\begin{array}{r}1 \\ -1\end{array}\right]\right)$ and $\left(2,\left[\begin{array}{r}1 \\ -2\end{array}\right]\right)$
- $\left(-2,\left[\begin{array}{r}1 \\ -1\end{array}\right]\right)$ and $\left(1,\left[\begin{array}{r}1 \\ -2\end{array}\right]\right)$
