Currently in the DE class I'm taking we're learning how to solve homogeneous linear systems. One of the steps requires you to come up with the associated eigenvector of a system. Unfortunately nowhere in our book does it discuss anything about eigenvectors and all our professor told us was that we'd find it easy if we had already taken linear algebra which I have not taken. I tried to find a good explanation of how to find eigenvectors of 2x2 and 3x3 matrices but didn't find much. For example how would I find the eigenvector of:

$\displaystyle \begin{vmatrix}

0 & -7 & 0\\

5 & 8 & 4\\

0 & 5 & 0

\end{vmatrix}

\begin{vmatrix}

k_{1}\\

k_{2}\\

k_{3}

\end{vmatrix}\Rightarrow

\begin{matrix}

-7k_{2}=0\\

5k_{1}+8k_{2}+4k_{3}=0\\

5k_{2}=0

\end{matrix}$

Thanks for your help.