<s>[INST] How to find the eigenvectors. $$\begin{vmatrix}
1&1&1&1\\
1&1&0&0\\
1&0&1&0\\
1&0&0&1\\
\end{vmatrix}$$
for this matrix the eigenvalues are $(1,1,1-\sqrt3,1+\sqrt 3)$
I if I try find the eigenvectors in the cases where the eigenvalue is $1$ I end up with two equations and $4$ unknowns. 
I'm not sure not to take the equivalent of the cross product for $4\times1$ matrices (I'm not sure it exists?) so how can I find the other two orthogonal eigenvectors? 
 [/INST] For $\lambda =1$
$$\begin{align} (A - I)v  = 0 &\implies \begin{pmatrix}0&1&1&1\\1&0&0&0\\1&0&0&0\\1&0&0&0\end{pmatrix}\begin{pmatrix}v_1\\v_2\\v_3\\v_4\end{pmatrix}= 0\\&\implies \begin{cases}v_1 = 0\\v_2 + v_3 + v_4 =0\end{cases}\end{align}$$
Then the eigenvectors associated to $\lambda =1 $ are of the form $$\begin{pmatrix}0\\-v_3-v_4\\v_3\\v_4\end{pmatrix}$$
Taking for example, $v_3 = 1$ and $v_4 = -1$, and then $v_3 = 1$ and $v_4 =2$ we have $$\begin{pmatrix}0\\0\\1\\-1\end{pmatrix} , \begin{pmatrix}0\\-3\\1\\2\end{pmatrix}$$
Notice that you may choose different values for $v_3$ and $v_4$. 
</s>