# 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 $\displaystyle (0,0)$, $\displaystyle (1,0)$ and $\displaystyle (-1,0)$.

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

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

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

and

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

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

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

but I can't get anywhere with it. I did give it a go and got $\displaystyle U_1=0$ 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 $\displaystyle \lambda$ = one of your $\displaystyle \lambda_1$ or $\displaystyle \lambda_2$, then I bring the RHS of your matrix equation to the left, I get:

\displaystyle \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=$\displaystyle \lambda$ .

3. To find eigenvectors always solve the system
$\displaystyle (A - \lambda I)v = 0.$
The result is the eigenvector $\displaystyle v$ corresponding to the eigenvalue $\displaystyle \lambda$.

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.

### eigenvectors of jacobian matrix

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