Hallo,
ν1=\( \begin{pmatrix} -1\\4\\0 \end{pmatrix} \)
allgemein gilt:
\( \begin{aligned} A \vec{v}_{1} & =\lambda \vec{v}_{1} \\ (A-\lambda E) \cdot \vec{v}_{2} & =\vec{v}_{1} \\ (A-\lambda E) \cdot \vec{v}_{3} & =\vec{v}_{2}\end{aligned} \)
\( \left(\begin{array}{ccc}0 & 0 & 1 \\ -8 & -2 & 0 \\ -4 & -1 & 2\end{array}\right)\left(\begin{array}{l}y1 \\ y2 \\ y3\end{array}\right)=\left(\begin{array}{c}-1 \\ 4 \\ 0\end{array}\right) \)
und:
\( \left(\begin{array}{ccc}0 & 0 & 1 \\ -8 & -2 & 0 \\ -4 & -1 & 2\end{array}\right)\left(\begin{array}{l}y1 \\ y2 \\ y3\end{array}\right)=\left(\begin{array}{c}-\frac{1}{2} \\ 0 \\ -1\end{array}\right) \)
Die Hauptvektoren lauten:
\( \mathrm{ν2}=\left(-\frac{1}{2}, 0,-1\right) \)
\( \mathrm{v3}=\left(0,0,-\frac{1}{2}\right) \)
Das Fundamentalsystem lautet:
\( \begin{array}{lr} \vec{y}_{1}(t)= & e^{\lambda t} \cdot \vec{v}_{1} \\ \vec{y}_{2}(t)= & e^{\lambda t}\left[\vec{v}_{2}+t \vec{v}_{1}\right] \\ \vec{y}_{3}(t)= & e^{\lambda t}\left[\vec{v}_{3}+t \vec{v}_{2}+\frac{t^{2}}{2} \vec{v}_{1}\right] \end{array} \)
