Proof $I_{n} - \frac{2}{(\lvert v \rvert)^{2}} * vv^{t}$ is an orthogonal matrix

Question

How do I proof that $I_{n} - \frac{2}{(\lvert v \rvert)^{2}} * vv^{t}$ is an orthogonal matrix, given that v $\in \mathbb{R}^{n}$ an non-trivial vector. I believe I could use the fact that $U * U^t = I_{n}$, but I do not really know.

Answer 1

Hint: For simplicity, define $u = v/|v|$, so that your matrix is $I - 2uu^\top$ and $|u| = 1$. Show that $(I - 2uu^\top)$ is a symmetric matrix, expand (i.e. "FOIL") the product $$ \left(I - 2uu^\top \right)\left(I - 2uu^\top \right), $$ and simplify using the fact that $uu^\top uu^\top = u(u^\top u)u^\top = uu^\top$.