For $\Bbb R^n$, Householder matrix $Q=I-2vv^T$ is an operator that maps a vector to its reflection across a hyperplane of normal $v$.
The following is an illustration for Householder operator of a general inner product space.
For $\Bbb R^n$, use standard inner product, then we have $x-2<x,v>v=x-2vv^Tx=(I-2vv^T)x$, where $I-2vv^T$ is the Householder matrix. It satisfies the following
However, I am wondering what would be the Householder matrix for complex vector space $\Bbb C^n$? It looks like $I-2vv^H$ is not right ($H$ denotes conjugate transpose), because by my calculation it does not satisfy above problem 5.7.3. Wikipedia suggests
I tried, and I might be wrong, but it does not seem to satisfy problem 5.7.3 either.
$Q(x+y)=(x + y) - \frac{{x - y}}{{{{\left\| {x - y} \right\|}^2}}}({(x - y)^H}(x + y) + \frac{{{x^H}(x - y)}}{{{{(x - y)}^H}x}}{(x - y)^H}(x + y))$
It looks impossible for $Q(x+y)=x+y$ unless $x^Hy=y^Hx$.