Suppose $q \in \mathbb{C}^m$ and $P$ is defined to be the orthogonal projection matrix onto the $range(q)$ i.e. $P=qq^*$. I am looking to find the SVD of the reflector matrix $I-2P$ where $I$ is the $m \times m$ identity matrix. This is what I have so far:
Suppose we take the full SVD of $P$ i.e. $P=U\Sigma V^*$. This tells us that $U = [q|Q^\perp]$ where $Q^\perp$ are the $m-1$ columns that are perpendicular to $\langle q \rangle$. Using the property of unitary matrices, we have that $I-2P=UU^*-2qq^*=qq^*+Q^\perp Q^{\perp^*}-2qq^*=Q^\perp Q^{\perp^*}-qq^*$.
I am not too sure where to go from here. I know that the singular values of this SVD are all 1, but I cannot seem to explicity show it.
Any and all help is greatly appreciated!