# How should I approach this multiple choice question?

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.

• Let $a$ be an arbitrary vector such that $v^Ta=0$. A straightforward calculation results in \eqalign{Pv&=-v\cr Pa&=a\cr}These are eigenvalue equations, and thus they answer (b) and (d). Multiplying these same vectors by $P^2$ will produce the answers to (a) and (c). – greg Jan 28 at 18:37
• $P$ is an orthogonal matrix ( see en.wikipedia.org/wiki/Householder_transformation), whence options (b) and (c) are correct. – StubbornAtom Jan 28 at 20:56

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. $$w^T v = 0$$.