# Fundamental systems

• Dec 8th 2010, 09:53 AM
LHeiner
Fundamental systems
Hello

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

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

Can somebody help me? I know it has something to do with Jordan normal form but my linear algebra is really worse (Crying).
THX very much!
• Dec 10th 2010, 02:11 AM
LHeiner
No idea? I really don't know where to start
• Dec 10th 2010, 02:36 AM
FernandoRevilla
Find $\displaystyle e^{tA}$, its columns provide a fundamental system.

In our case,

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

so,

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

Fernando Revilla
• Dec 10th 2010, 04:30 AM
LHeiner
I'm sorry but what do you mean by $\displaystyle A=\textrm{diag}(A_1,A_2)$?
• Dec 10th 2010, 10:52 AM
FernandoRevilla
Quote:

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

Diagonal block matrix.

Fernando Revilla
• Dec 10th 2010, 11:36 AM
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:

$\displaystyle A=\begin{pmatrix} 2 &1 &0 & 0 & 0\\ 0& 2 &0 & 0 & 0\\ 0& -1 &2 &0 & 0\\ 0 & 0 & 0 & 2 &2 \\ 0& 0 & 0 & -2 &0 \end{pmatrix} \rightarrow \begin{pmatrix} 2 &0 &0 & 0 & 0\\ 0& 2 &0 & 0 & 0\\ 0& 0 &2 &0 & 0\\ 0 & 0 & 0 & 2 & 0\\ 0& 0 & 0 & 0 &2 \end{pmatrix}$
• Dec 10th 2010, 11:44 AM
TheEmptySet
Quote:

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:

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

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.