Solution
= Solution
For every <vector> $v$, the <Hermitian inner product> gives
$$
\|Cv\|^2=v^\dagger C^\dagger Cv=0.
$$
The <norm> vanishes only on the zero <vector>, so $Cv=0$ for every $v$ and $\boxed{C=0}$. In contrast,
$$
\boxed{N=\begin{pmatrix}0&1\\0&0\end{pmatrix}\ne0,\qquad N^2=0}
$$
is a nonzero <nilpotent matrix>. Multiplying a <matrix> by its <conjugate transpose> detects column <norms>; multiplying it by itself has no corresponding positivity property.