Let $v\in\mathbb{R}^k$ with $v^Tv\neq 0$. Let $$P=I-2\frac{vv^T}{v^Tv}$$ where $I$ is the $k\times k\ $ identity matrix. Then which of the following statements is (are) true?
(a) $P^{-1}=I-P$
(b) $-1$ and $1$ are eigenvalues of $P$
(c) $P^{-1}=P$
(d) $(I+P)v= v$
I'd really appreciate it if someone could just run me through the process of checking each option while briefly touching upon the relevant properties.
For (a) and (c) you can check if $P P^{-1} = I$. For (d) try to verify by direct computation. For (b) try to check the eigenvalues for the eigenvectors $v$ and $w$ where $w$ is orthogonal to $v$, i.e. $w^T v = 0$.