Linear algebra: orthonormal eigenvectors Let $A=\begin{pmatrix} 1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \end{pmatrix}$
Its eigenvalues are $\lambda=0,0,3$.
Find two pairs of orthonormal eigenvectors for $\lambda=0$ and
show that $P=x_1 x_1^{T}+x_2 x_2^{T}$ is the same for both pairs.
(These eigenvectors make orthogonal matrix $Q$, and $A=Q\lambda Q^{-1}$) 
I found first pair $x_1=\frac{1}{\sqrt{2}}(-1,1,0), x_2=\frac{1}{\sqrt{2}}(-1,0,1)$
and second pair $x_1=\frac{1}{\sqrt{6}}(2,-1,-1), x_2=\frac{1}{\sqrt{2}}(0,1,-1)$
but then $P$ of these two pairs are not same.
Where am I wrong?  
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I solve it. 
Put  $x_1=\frac{1}{\sqrt{2}}(-1,1,0), x_2=\frac{1}{\sqrt{6}}(1,1,-2)$ in the first pair.
I'm not sure there is any particular formula for this question,
but I just put all possible eigenvectors here.
Thanks for your answers and comments. 

The first pair is not orthonormal. The inner product is $1/2$.
Since any nonzero constant multiple of an eigenvector is also an eigenvector, we take $x_1=(-1,1,0)$ and $x_2=(-1,0,1)$. Then $\langle x_1,x_2 \rangle = -1\cdot -1 + 1 \cdot 0 + 0 \cdot 1 = 1 \neq 0$.
