Item8
For any vector \(q\text{,}\) with \(\|q\|_2=1\text{,}\) we can define the projectors
\begin{equation*}
P_{q}=qq^\star \qquad \text{ and } \qquad
P_{\perp q} = I - q q^\star\text{.}
\end{equation*}
More generally, for any vector
\begin{equation*}
P_a=\frac{aa^\star}{a^\star a}\text{.}
\end{equation*}
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