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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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