EDIT: just writing out my working. 2 mins.
http://www.wolframalpha.com/input/?i=eigenvalues+{{1%2C+1%2C+1}%2C{2%2C2%2C+2}%2C+{3 %2C+3%2C+3}}
How is {1, 2, 3} and answer(Corresponding to EV = 6)? I get {2, 1, 0) and just can't seem to work out how they get (1, 2, 3)!
Eigen Value 6:
$\displaystyle \begin{matrix}5 & -1 & -1 \\ -2 & 4 & -2 \\ -3 & 3 & -3 \end{matrix}$
So, for
$\displaystyle -3x -3y + 3z = 0$
$\displaystyle x + y - z = 0$
I obtain:
$\displaystyle y\vec{-1, 1, 0) + z\vec{1, 0 , 1)$
$\displaystyle -2x -4y + -2z = 0$
$\displaystyle x -2y + z = 0$
I obtain:
$\displaystyle y\vec{2, 1, 0) + z\vec{-1, 0 , 1)$
...And actually just typing this out i've mega confused myself as there are now 4 different basis and there's meant to be a max of 3 :S.