<s>[INST] Help finding eigenvectors? The given matrix is:
$$
\begin{pmatrix}3 & 1 & 6 \\ 2 & 1 & 0 \\ -1 & 0 & -3\end{pmatrix}\qquad
$$
I got the characteristic polynomial of $$x^3 - x^2 - 5x - 3 = 0$$
which factors down to $$(x+1)^2 * (x-3) = 0$$
I see that it  has eigenvalues of -1 and 3.
I know I'm almost there, I plugged in the eigenvalues to $A-\lambda I$ but completely forgot how to find the eigenvectors after this.
When $\lambda$ = 3, I got:
$$\begin{pmatrix}0 & 1 & 6 \\ 2 & -2 & 0 \\ -1 & 0 & 0\end{pmatrix} \begin{pmatrix}x
_1 \\ x_2 \\ x_3\end{pmatrix}=0\qquad$$
and when $\lambda$ = -1, I got:
$$\begin{pmatrix}4 & 1 & 6 \\ 2 & 0 & 0 \\ -1 & 0 & 4\end{pmatrix}\begin{pmatrix}x
_1 \\ x_2 \\ x_3\end{pmatrix}=0\qquad$$
Where do I go from here? Row-reduce the $3x3$ matrices to solve?
 [/INST] it is enough you find:
$$\left(\begin{bmatrix}3 & 1 & 6 \\ 2 & 1 & 0 \\ -1 & 0 & -3\end{bmatrix}- 3I\right)\begin{bmatrix}x
_1 \\ x_2 \\ x_3\end{bmatrix}=0\qquad
\text{and} \qquad 
\left(\begin{bmatrix}3 & 1 & 6 \\ 2 & 1 & 0 \\ -1 & 0 & -3\end{bmatrix}+ I\right)\begin{bmatrix}x
_1 \\ x_2 \\ x_3\end{bmatrix}=0 \ .$$
</s>