Toronto Math Forum
MAT2442013S => MAT244 MathTests => Quiz 4 => Topic started by: Victor Ivrii on March 22, 2013, 04:18:05 AM

Please post the problem and its solution (in contrast to a popular opinion I am not omniscient :D and unless someone advises me I am not sure which problem it was).

I think the first problem was from the textbook
$
\mathbf x' = \left(\begin{array}{ccc}
1 & 1 & 1 \\
2 & 1 & 1 \\
0 & 1 & 1 \\
\end{array}\right) \mathbf x \\
$
The characteristic polynomial is
$
4 + 3k^2  k^3 = 0 \\
k = 2, 2, 1
$
The eigenvectors are
$
\lambda = 1,
\left(\begin{array}{c}
3 \\
4 \\
2
\end{array}\right),
\lambda = 2,
\left(\begin{array}{c}
0 \\
1 \\
1
\end{array}\right)
$
Jordan decomposition yields the similarity transform of
$
\mathbf T = \left(\begin{array}{ccc}
3 & 0 & 1 \\
4 & 1 & 1 \\
2 & 1 & 0
\end{array}\right)
$
Thus the solution is
$
\mathbf x = c_1 \left(\begin{array}{c} 3 \\ 4 \\ 2 \end{array}\right) e^{t} + c_2 \left(\begin{array}{c} 0 \\ 1 \\ 1 \end{array}\right) e^{2t} + c_3 \left( \left(\begin{array}{c} 0 \\ 1 \\ 1 \end{array}\right) t e^{2t} + \left(\begin{array}{c} 1 \\ 1 \\ 0 \end{array}\right) e^{2t}\right)
$