Fundamental systems

**LHeiner** · December 8th 2010, 09:53 AM

Hello

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

$ x'=\begin{pmatrix} 2 &1 &0 & 0 & 0\\ 0& 2 &0 & 0 & 0\\ 0& -1 &2 &0 & 0\\ 0 & 0 & 0 & 2 &2 \\ 0& 0 & 0 & -2 &0 \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!

**LHeiner** · December 10th 2010, 02:11 AM

No idea? I really don't know where to start

**FernandoRevilla** · December 10th 2010, 02:36 AM

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

**LHeiner** · December 10th 2010, 04:30 AM

I'm sorry but what do you mean by $A=\textrm{diag}(A_1,A_2)$ ?

**FernandoRevilla** · December 10th 2010, 10:52 AM

Originally Posted by LHeiner

I'm sorry but what do you mean by $A=\textrm{diag}(A_1,A_2)$ ?

Diagonal block matrix.

Fernando Revilla

**LHeiner** · December 10th 2010, 11:36 AM

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:

$A=\begin{pmatrix} 2 &1 &0 & 0 & 0\\ 0& 2 &0 & 0 & 0\\ 0& -1 &2 &0 & 0\\ 0 & 0 & 0 & 2 &2 \\ 0& 0 & 0 & -2 &0 \end{pmatrix} \rightarrow \begin{pmatrix} 2 &0 &0 & 0 & 0\\ 0& 2 &0 & 0 & 0\\ 0& 0 &2 &0 & 0\\ 0 & 0 & 0 & 2 & 0\\ 0& 0 & 0 & 0 &2 \end{pmatrix}$

**TheEmptySet** · December 10th 2010, 11:44 AM

Originally Posted by LHeiner

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:

$A=\begin{pmatrix} 2 &1 &0 & 0 & 0\\ 0& 2 &0 & 0 & 0\\ 0& -1 &2 &0 & 0\\ 0 & 0 & 0 & 2 &2 \\ 0& 0 & 0 & -2 &0 \end{pmatrix} \rightarrow \begin{pmatrix} 2 &0 &0 & 0 & 0\\ 0& 2 &0 & 0 & 0\\ 0& 0 &2 &0 & 0\\ 0 & 0 & 0 & 2 & 0\\ 0& 0 & 0 & 0 &2 \end{pmatrix}$

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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