= Reflection length of a Euclidean orthogonal map
{title2=$\ell(Q)=\operatorname{rank}(I-Q)$}
A product of $m$ <reflections> has $\operatorname{rank}(I-Q)\le m$, by the rank inequality applied successively to $I-AB=(I-A)+A(I-B)$. Conversely an <orthogonal transformation> fixes the orthogonal complement of its moved space; successive Householder reflections fix one new basis direction at each step in that moved space. At most its dimension $\operatorname{rank}(I-Q)$ are required. Thus central inversion in three dimensions needs exactly three reflections.
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