I am given the following matrix:
$A = P\left(\begin{matrix} -5 & -6 & 3\\3 & 4 & -3\\0 & 0 & -2\end{matrix}\right)$
After finding the following eigenvalues by finding the characteristic polynomial I get:
$\lambda_1 = \lambda_2 = -2$ and $\lambda_3 = 1$
After finding the I now need to find the eigenvectors for $\lambda_1$ and $\lambda_2$. After putting matrix into reduced-row echelon form for $\lambda_1$:
$\left(\begin{matrix} 1 & 2 & -1\\0 & 0 & 0\\0 & 0 & 0\end{matrix}\right)$
I now want to find the eigenvector from this, but am I bit puzzled how to find it an then find the basis for the eigenspace (I know this involves putting it into vector form, but for some reason I found the steps to translating-to-vector-form really confusing and still do).
A step-by-step explanation on this point would be very helpful for a linear algebra newbie.