I am studying the following Householder reflection in $\mathbb{R}^2$:
$H = I - 2 \frac{vv^t}{v^tv}$, with $v = x - y$ the result of subtracting any two different and non-zero vectors in $\mathbb{R}^2$.
I can see geometrically and by taking some examples that it seems to happen that $Hx = y$ and $Hy = x$, but I can't seem to be able to prove it for any pair of $x$ and $y$.
Does that property hold?