<s>[INST] What are the Eigenvectors in the following matrix? I have the matrix A:
\begin{bmatrix}
4 & 2 & 2\\
2 & 4 & 2\\
2 & 2 & 4\\
\end{bmatrix}
I found $\lambda I_n - A$ to be:
\begin{bmatrix}
(\lambda -4) & -2 & -2\\
-2 & (\lambda -4) & -2\\
-2 & -2 & (\lambda -4)\\
\end{bmatrix}
Applying Sarrus' Rule, and with help in a previous question, I determined the Eigenvalues to be $\lambda = 8$ or $2$ from the polynomial $\lambda^3 -12\lambda^2 +36\lambda -32 = 0$
If I now get the matrix
\begin{bmatrix}
-2 & -2 & -2\\
-2 & -2 & -2\\
-2 & -2 & -2\\
\end{bmatrix} when verifying $\lambda = 2$, does this mean that $\lambda = 2$ is not a valid Eigenvalue? Or does it mean that $(\lambda I_n -A)v = 0$
Vector v = 1? And creates an Eigenspace from this?
Checking for $\lambda = 8$ gives me:
\begin{bmatrix}
6 & -2 & -2\\
-2 & 6 & -2\\
-2 & -2 & 6\\
\end{bmatrix}
How do I get the Eigenvectors for this?
 [/INST] An easier way to check for eigenvalues is to factorize
$$
x^3 - 12x^2 + 36x -32 = \left(x-8\right)\left(x-2\right)^2
$$
Then to get the eigenvectors, input the eigenvalues one by one to
$$
(\lambda I - A) \cdot \begin{bmatrix}S_1\\S_2\\S_3\end{bmatrix} = \begin{bmatrix}0\\0\\0\end{bmatrix}
$$
and solve for $S_1$, $S_2$ and $S_3$
</s>