- #1
Somefantastik
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[tex] A = \[ \left( \begin{array}{ccc}
0 & 0 & 1 \\
0 & 1 & 0 \\
1 & 0 & 0 \end{array} \right)\] [/tex]
eigenvalues are [tex] \lambda_{1} = -1, \ \lambda_{2} = \lambda_{3} = 1 [/tex]
[tex](A-\lambda_{1}I)u^{(1)} = 0 \ => \ u^{(1)} = \[ \left( \begin{array}{c}
-1 \\
0 \\
1 \end{array} \right)\] [/tex]
[tex] (A-\lambda_{2}I)u^{(2)} = 0 \ => u^{(1)} = \[ \left( \begin{array}{c}
1 \\
0 \\
1 \end{array} \right)\] [/tex]
[tex] (A -\lambda_{3}I)u^{(3)} = u^{(2)} \ => \
\left(
\begin{array}{ccc|c}
1&0&-1&-1\\
0&0 &0&1\\
0&0&0&0
\end{array}
\right)
[/tex]
since we cannot have 0 = 1, we can say that there is only one eigenvector for
[tex]\lambda = 1 [/tex]
which means that the Jordan form will be
[tex] \[ \left( \begin{array}{ccc}
1 & 1 & 0 \\
0 & 1 & 0 \\
0 & 0 & -1 \end{array} \right)\] [/tex]
Am I correct?
Now I need to find exp(Jt) and I'm not sure how.
If I only have 2 eigenvectors, how can I find the fund. matrix T such that
[tex]e^{At} = Te^{Jt}T^{-1} [/tex]?
0 & 0 & 1 \\
0 & 1 & 0 \\
1 & 0 & 0 \end{array} \right)\] [/tex]
eigenvalues are [tex] \lambda_{1} = -1, \ \lambda_{2} = \lambda_{3} = 1 [/tex]
[tex](A-\lambda_{1}I)u^{(1)} = 0 \ => \ u^{(1)} = \[ \left( \begin{array}{c}
-1 \\
0 \\
1 \end{array} \right)\] [/tex]
[tex] (A-\lambda_{2}I)u^{(2)} = 0 \ => u^{(1)} = \[ \left( \begin{array}{c}
1 \\
0 \\
1 \end{array} \right)\] [/tex]
[tex] (A -\lambda_{3}I)u^{(3)} = u^{(2)} \ => \
\left(
\begin{array}{ccc|c}
1&0&-1&-1\\
0&0 &0&1\\
0&0&0&0
\end{array}
\right)
[/tex]
since we cannot have 0 = 1, we can say that there is only one eigenvector for
[tex]\lambda = 1 [/tex]
which means that the Jordan form will be
[tex] \[ \left( \begin{array}{ccc}
1 & 1 & 0 \\
0 & 1 & 0 \\
0 & 0 & -1 \end{array} \right)\] [/tex]
Am I correct?
Now I need to find exp(Jt) and I'm not sure how.
If I only have 2 eigenvectors, how can I find the fund. matrix T such that
[tex]e^{At} = Te^{Jt}T^{-1} [/tex]?
Last edited: