I'm trying to compute the eigenvector of the following matrix:
$$ A = \begin{bmatrix} 0.3889 & 0.3456 \\ 0.3456 & 0.4044 \\ \end{bmatrix} $$
where one of the eigenvalues is $\lambda=0.7423$. This example was taken from my course and I only have the solution:
$$\vec{v}=\begin{bmatrix} 0.6991\\ 0.7150\\ \end{bmatrix}$$
The solution is correct since $A\vec{v}=\lambda\vec{v}$, but I don't understand how to find it. The system of equations I'd have to solve would be:
$$-0.3534v_{1}+0.3456v_{2}=0\\0.3456v_{1}-0.3379v_{2}=0$$
The solution is clearly $0$ for both components. What am I missing here?
I think you're missing that there are solutions other than $v_1=v_2=0$
If $-.3534v_1+.3456v_2=0$, then $.3456v_2=.3534v_1$, or $v_2=1.022569v_1$. So $\left(\begin{matrix}1\\1.022569\end{matrix}\right)$ is a solution, as well as $\left(\begin{matrix}1*.6991\\1.022569*.6991\end{matrix}\right)=\left(\begin{matrix}.6991\\.7149\end{matrix}\right)$