Solution
= Solution
If $u\cdot v=0$, use the positively oriented <orthonormal basis> $(u,v\times u,v)$. The <cross product> identities give $Au=u+v\times u$, $A(v\times u)=-u$ and $Av=0$. In this basis,
$$
A=\begin{pmatrix}1&-1&0\\1&0&0\\0&0&0\end{pmatrix}.
$$
Its <characteristic polynomial> is $\lambda(\lambda^2-\lambda+1)$, so
$$
\boxed{\lambda=0,\quad\frac{1+i\sqrt3}{2},\quad\frac{1-i\sqrt3}{2}.}
$$