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Consider a linear system with state space representation is x and (t) = Ax (t). If the initial state vector of the system is x (0) =$ \left[ \begin{array} \ 1 \\ -2 \end{array} \right] $, the system response is x (t) = $ \left( \begin{array} \ e^{-2t} \\ -2 e^{-2t} \end{array} \right) $. If the initial state vector of the system changes, the system response becomes x(t) = $ \left[ \begin{array} \ e^{-t} \\

  • e^{-6} \end{array} \right] $.

The eigen value and eigen vector pairs ($\lambda_i, V_i$) for the system are

  1. $ \left( \begin{array} \ -1, & \left[ \begin{array} \ 1 \\\\ -1 \end{array} \right] \end{array} \right) $and $ \left( \begin{array} \ -2, & \left[ \begin{array} \ 1 \\\\ -2 \end{array} \right] \end{array} \right) $
  2. $ \left( \begin{array} \ -1, & \left[ \begin{array} \ 1 \\\\ -1 \end{array} \right] \end{array} \right) $and $ \left( \begin{array} \ 2, & \left[ \begin{array} \ 1 \\\\ -2 \end{array} \right] \end{array} \right) $
  3. $ \left( \begin{array} \ 1, & \left[ \begin{array} \ 1 \\\\ -1 \end{array} \right] \end{array} \right) $and $ \left( \begin{array} \ -2, & \left[ \begin{array} \ 1 \\\\ -2 \end{array} \right] \end{array} \right) $
  4. $ \left( \begin{array} \ -2, & \left[ \begin{array} \ 1 \\\\ -1 \end{array} \right] \end{array} \right) $and $ \left( \begin{array} \ -1, & \left[ \begin{array} \ 1 \\\\ -2 \end{array} \right] \end{array} \right) $
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