MAT244-2014F > Quiz 4

Q4 problem 1

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Shuyang Wang:
\begin{equation*} \textbf{x}'=\begin{pmatrix} -2 & \hphantom{-}1\\  \hphantom{-}1 &-2 \end{pmatrix}\textbf{x}\ . \end{equation*}
find eigenvalues
\begin{equation*} \det (A - rI) = \left|\begin{matrix}-2 - r &1\\1&  - 2 - r\end{matrix}\right| =  r^2+ 4r + 3 = 0\implies r_1=-3, r_2=-1\end{equation*}
then, find eigenvectors
\begin{equation*} \begin{pmatrix} -2 - r & \hphantom{-}1\\  \hphantom{-}1 &-2 -r\end{pmatrix}\begin{pmatrix}\mathbf{\xi}_1\\\mathbf{\xi}_2\end{pmatrix}=\begin{pmatrix}0\\0\end{pmatrix} \end{equation*}
then
\begin{equation*}\mathbf{\xi}^1 =\begin{pmatrix}1\\-1\end{pmatrix} , \mathbf{\xi}^2 =\begin{pmatrix}1\\1\end{pmatrix}\end{equation*}
so
\begin{equation*}\mathbf{x}= C_1e^{-3t}\begin{pmatrix}1\\-1\end{pmatrix}+ C_2e^{-t}\begin{pmatrix}1\\1\end{pmatrix}\end{equation*}
plot(stable):

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