<s>[INST] Help with finding eigenvectors and polynomial factorization I have a real and invertible matrix
$$ A = \left( \begin{matrix} 5 & 4 \\
  4&5 \end{matrix} \right)$$
and I need to find the eigenvectors.
The characteristic polynomial is $(5-\lambda)^2 - 16$. According to Wolfram Alpha this factors to $(1-\lambda)(9-\lambda)$ (How does one get there, are there any good tutorials on polynomial factorization?), so the eigenvalues are $1$ and $9$.
Calculating the eigenvector to the eigenvalue $1$ given by $(A-I)\vec x = \vec 0$.
I get that $\vec x$ is the null vector $=\left(
\begin{array}{c}
0\\
0\\
\end{array}
\right)$(can that be right?).
For the second eigenvector $(A-9I)\vec x = 0$ I get 
$$x_{1} = 4x_{2}$$
$$-x_{2} = 5x_{1}$$
Is this right, and how can I write the second eigenvector with this information??
 [/INST] Well $(5-\lambda)^2-4^2$ is a difference of squares, but you really need to learn factoring.
$$\begin{pmatrix}
4&4\\4&4\\\end{pmatrix}\begin{pmatrix}x\\y\\\end{pmatrix}=0$$
gives $x=-y$ so $\begin{pmatrix}1\\-1\\\end{pmatrix}$ is one eigenvector,
and $$\begin{pmatrix}
-4&4\\4&-4\\\end{pmatrix}\begin{pmatrix}x\\y\\\end{pmatrix}=0$$
gives $x=y$ so $\begin{pmatrix}1\\1\\\end{pmatrix}$ is the other eigenvector,
</s>