do Householder reflections describe all reflections?

Question

I'm familiar with Householder reflections; they are a simple transformation that, given a normal vector, describes reflection in the hyperplane perpendicular to that vector.

But do Householder reflections describe the complete space of matrices describing reflection about the origin, or are there other reflections that cannot be described by Householder reflections?

Answer 1

Mariano's answer should be enough, if you read your link to Wikipedia carefully, but maybe I could add some hints.

If you want to do a reflection about a hyperplane containing the origin, you pick a unitary orthogonal vector to that hyperplane, $v$, and then write the vector $x$ you want to reflect about $[v]^\bot$ as a sum

$$ x = \lambda v + u $$

with $u \in [v]^\bot$. Now, you want to determine $\lambda \in \mathbb{R}$ in order this to be true:

$$ v \cdot x = \lambda v\cdot v + v\cdot u = \lambda \ . $$

So, necessarily

$$ x = (v\cdot x) v + u \ . $$

Hence, if you want to reflect $x$ about $[v]^\bot$, you just substract two times its $v$-component:

$$ P_v x = x - 2 (v\cdot x) v = x - 2 v (v^t x) = (I - 2vv^t ) x \ . $$

So any (orthogonal) reflexion about any hyperplane $[v]^\bot$ is of the form

$$ P_v = I - 2vv^t \ . $$

That is, a Housholder reflection.

Answer 2

Yes, they are all reflections.

Answer 3

An additional consideration to this is whether "reflection" means reflection in the hyperplane of dimension $n-1$, where $n$ is the dimension of the containing space, or other kinds of reflection, such as reflection through a point. If you include reflection through a point, Householder reflections do not contain them all.