Multiple eigenvectors for an eigenvalue and how to know We have the following matrix:
$$\begin{pmatrix}
    0 & -1 & 0 \\
    4 & 4 & 0 \\
    2 & 1 & 2
\end{pmatrix} $$
This matrix has characteristic equation $- \lambda ^3 + 6 \lambda ^2 - 12 \lambda + 8$, which gives us the eigenvalue $\lambda = 2$. However, this eigenvalue has two eigenvectors, namely $(1,-2,0)$ and $(0,0,1)$. 
How would you mathematically be able to know how many eigenvectors are associated with a single eigenvector? Is there a way to find out? 

Yes let $v=(x,y,z)^T$ an eigenvector for the given matrix $A$ associated to $2$ then we have
$$Av=2v$$
and if we solve the system of equations we find that $v$ is a linear combination of the two vectors given in your question.
