Solution
= Solution
The three eigenvalues are
$$
1,\qquad e^{i\pi/4}=\frac{1+i}{\sqrt2},
\qquad e^{3i\pi/4}=\frac{-1+i}{\sqrt2}.
$$
Thus $N(U)$ is the filled triangle with vertices
$$
(1,0),\qquad(2^{-1/2},2^{-1/2}),
\qquad(-2^{-1/2},2^{-1/2})
$$
in the complex plane. Its upper edge is the horizontal chord joining the last two vertices, and the other two edges join those points to $1$. In particular, the origin lies outside this triangle.