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Jordan Form and Linear Systems: Standard Exercisesmd 7e17606
exercise/math/differential-equations/jordan-form-and-linear-systems.exercise.n.md
Jordan Form and Linear Systems: Standard Exercises
mathdifferential-equationsexercisejordan-formlinear-systems
1Problem 1: A Jordan Block of Size Three
For
J=\begin{pmatrix}2&1&0\\0&2&1\\0&0&2\end{pmatrix}=2I+N,
compute N^2,N^3 and e^{Jt}. Also write the solution satisfying \boldsymbol{x}(0)=e_3.
1.1Sample Answer
○
N^2=\begin{pmatrix}0&0&1\\0&0&0\\0&0&0\end{pmatrix},\qquad N^3=0,
e^{Jt}=e^{2t}
\begin{pmatrix}
1&t&t^2/2\\
0&1&t\\
0&0&1
\end{pmatrix}.
Therefore,
\boldsymbol{x}(t)=e^{2t}\begin{pmatrix}t^2/2\\t\\1\end{pmatrix}.