Similarity Mapping on Plane with Negative Parameter

Theorem
Let $\beta \in \R_{<0}$ be a (strictly) negative real number.

Let $s_\beta: \R^2 \to \R^2$ be the similarity mapping on $\R^2$ whose scale factor is $\beta$.

If $\beta < 0$, then $s_\beta$ is a stretching or contraction followed by a rotation one half turn.

It is also the same as a rotation one half turn followed by a stretching or contraction.