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Math Help - Fundamental systems

  1. #1
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    Fundamental systems

    Hello

    Once again I'm stuck: I need to calculate (explicitly) a (real) fundamental system of the following ODE:

    <br />
x'=\begin{pmatrix}<br />
2 &1  &0  & 0 & 0\\ <br />
 0& 2 &0  & 0 & 0\\ <br />
 0& -1 &2  &0  & 0\\ <br />
0 & 0 & 0 & 2 &2 \\ <br />
 0& 0 & 0 & -2 &0 <br />
\end{pmatrix} x


    Can somebody help me? I know it has something to do with Jordan normal form but my linear algebra is really worse .
    THX very much!
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  2. #2
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    No idea? I really don't know where to start
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  3. #3
    MHF Contributor FernandoRevilla's Avatar
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    Find e^{tA}, its columns provide a fundamental system.

    In our case,

    A=\textrm{diag}(A_1,A_2)

    so,

    e^{tA}=\textrm{diag}(e^{tA_1},e^{tA_2})

    Fernando Revilla
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  4. #4
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    I'm sorry but what do you mean by A=\textrm{diag}(A_1,A_2)?
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  5. #5
    MHF Contributor FernandoRevilla's Avatar
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    Quote Originally Posted by LHeiner View Post
    I'm sorry but what do you mean by A=\textrm{diag}(A_1,A_2)?
    Diagonal block matrix.

    Fernando Revilla
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  6. #6
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    Mhh, i really can't see the blocks, i can transform the matrix into a diagonal matrix but is this allowed in this case and does that help me:

    A=\begin{pmatrix}<br />
2 &1 &0 & 0 & 0\\<br />
0& 2 &0 & 0 & 0\\<br />
0& -1 &2 &0 & 0\\<br />
0 & 0 & 0 & 2 &2 \\<br />
0& 0 & 0 & -2 &0<br />
\end{pmatrix}  \rightarrow \begin{pmatrix}<br />
2 &0 &0 & 0 & 0\\<br />
0& 2 &0 & 0 & 0\\<br />
0& 0 &2 &0 & 0\\<br />
0 & 0 & 0 & 2 & 0\\<br />
0& 0 & 0 & 0 &2<br />
\end{pmatrix}
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  7. #7
    Behold, the power of SARDINES!
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    Quote Originally Posted by LHeiner View Post
    Mhh, i really can't see the blocks, i can transform the matrix into a diagonal matrix but is this allowed in this case and does that help me:

    A=\begin{pmatrix}<br />
2 &1 &0 & 0 & 0\\<br />
0& 2 &0 & 0 & 0\\<br />
0& -1 &2 &0 & 0\\<br />
0 & 0 & 0 & 2 &2 \\<br />
0& 0 & 0 & -2 &0<br />
\end{pmatrix}  \rightarrow \begin{pmatrix}<br />
2 &0 &0 & 0 & 0\\<br />
0& 2 &0 & 0 & 0\\<br />
0& 0 &2 &0 & 0\\<br />
0 & 0 & 0 & 2 & 0\\<br />
0& 0 & 0 & 0 &2<br />
\end{pmatrix}
    You need to find the characteristic polynomial to find the eigenvalues. After you have the eigenvalues you can find a basis of genervalized eigenvectors, from there you will know the size of each Jordan block for each eigenvalue, so you can write down the Jordan Normal form.
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