# Thread: Eigenvectors of a Jacobian matrix

1. ## Eigenvectors of a Jacobian matrix

Sorry, if this is the wrong subforum, firstly.

So I have a dynamical system with fixed points $(0,0)$, $(1,0)$ and $(-1,0)$.

The bit I'm stuck on is this; I have the Jacobian matrix for $(1,0)$ and $(-1,0)$ as $A=\left( \begin{array}{cc} 0 & 1 \\ -2 & 1 \end{array}\right)$ (so the trace $\tau=1$ and the determinant $\delta=2$) and the eigenvalues as $\lambda_1=\frac{1}{2}(1_+7i)$ and $\lambda_2=\frac{1}{2}(1-7i)$.

To find the eigenvectors $\mathbf{U}$ and $\mathbf{V}$ I know I need to solve

$A=\left( \begin{array}{cc} 0 & 1 \\ -2 & 1 \end{array}\right) \left( \begin{array}{c} U_1 \\ U_2 \end{array} \right)=\frac{1}{2}(1+7i)\left( \begin{array}{c} U_1 \\ U_2 \end{array} \right)$

and

$A=\left( \begin{array}{cc} 0 & 1 \\ -2 & 1 \end{array}\right) \left( \begin{array}{c} V_1 \\ V_2 \end{array} \right)=\frac{1}{2}(1-7i)\left( \begin{array}{c} V_1 \\ V_2 \end{array} \right)$

and this is what I'm stuck on. I've found both lines of the simultaneous equations to be

$U_2=\frac{1}{2}(1+7i)U_1$
$-2U_1+U_2=\frac{1}{2}(1+7i)U_2$

but I can't get anywhere with it. I did give it a go and got $U_1=0$ but I get the feeling that's not right this was never my strong point when we initially learned it, but I think it's the complex number that's confusing me. Any ideas? Cheers in advance.

2. ## Finding the eigenvector

This seems to be an eigenvector problem. If I call $\lambda$ = one of your $\lambda_1$ or $\lambda_2$, then I bring the RHS of your matrix equation to the left, I get:


\left[
\begin {array}{ccc}
{0-\lambda}&1\\
\noalign{\medskip}
-2&{1-\lambda}
\end {array}
\right]
\left[
\begin {array}{ccc}
U1\\
\noalign{\medskip}
U2
\end {array}
\right]=
\left[
\begin {array}{ccc}
0\\
\noalign{\medskip}
0
\end {array}
\right]

I can then choose U1=1 and U2= $\lambda$ .

3. To find eigenvectors always solve the system
$
(A - \lambda I)v = 0.
$

The result is the eigenvector $v$ corresponding to the eigenvalue $\lambda$.

4. See, I knew all of this, but that wasn't what the problem was, it was the fact there was complex numbers there, I needed walking through it a little. S'alright though, I handed my thing in now, so we'll see how I did.

5. The complex numbers should just be treated as any scalar. Sorry, I didn't understand completely your question.