Statement for Linked Answer Questions 80 and 81: Consider a linear system whose state-space representation is $\dot{x}(t)=Ax(t)$. If the initial state vector of the system is $x(0)=[1;-2]$, then the system response is $x(t)=[e^{-2t};-2e^{-2t}]$. If the initial state vector changes to $x(0)

Question

Statement for Linked Answer Questions 80 and 81: Consider a linear system whose state-space representation is $\dot{x}(t)=Ax(t)$. If the initial state vector of the system is $x(0)=[1;-2]$, then the system response is $x(t)=[e^{-2t};-2e^{-2t}]$. If the initial state vector changes to $x(0)=[1;-1]$, then the system response becomes $x(t)=[e^{-t};-e^{-t}]$. The eigenvalue and eigenvector pairs $(\lambda,v)$ for the system are

Accepted Answer

(-1,[1;-1]) and (-2,[1;-2]). For a linear system, if $x(0)=v$ and the response is $x(t)=e^{\lambda t}v$, then $v$ is an eigenvector and $\lambda$ is the corresponding eigenvalue. Here, $[1;-2]$ produces $e^{-2t}[1;-2]$, so $(\lambda,v)=(-2,[1;-2])$. Similarly, $[1;-1]$ produces $e^{-t}[1;-1]$, so the other pair is $(-1,[1;-1])$.

Solution

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