Difference between these two eigenvectors? i have a simple question regarding eigenvalues and eigenvectors. Consider the following matrix:
$\left(\begin{array}{ccc}
1 & 1 & 3 \\
1 & 5 & 1 \\
3 & 1 & 1 \end{array}\right)$
The eigenvalues are $\lambda = -2,3,6$
Supposedly the respective eigenvectors are:
$\frac{1}{\sqrt{2}}\left(\begin{array}{c}
-1 \\
0 \\
1 \end{array}\right)$, 
$\frac{1}{\sqrt{3}}\left(\begin{array}{c}
1 \\
-1 \\
1 \end{array}\right)$, and 
$\frac{1}{\sqrt{6}}\left(\begin{array}{c}
1 \\
2 \\
1 \end{array}\right)$.
Would the solution be wrong if i flipped the signs in the first eigenvector? That is, would an eigenvector of 1/$\sqrt{2} \left\{1; 0; -1\right\}$ be wrong and/or change the values of the other eigenvectors?

No, that is a valid eigenvector too. Adding a sign to one of the eigenvectors does not change the others. 
