= Householder factorization of an orthogonal matrix
{c}
{title2=$Q=H_1\cdots H_r,\quad r\leq n$}
Every real <orthogonal matrix> of dimension $n$ is a product of at most $n$ <Householder matrices>. For a first-column unit vector $q\ne e_1$, the reflection with normal $q-e_1$ sends $q$ to $e_1$. Orthogonality then makes the transformed matrix $\operatorname{diag}(1,Q')$, reducing the problem by one dimension. If $q=e_1$, omit that reflection. The dimension-one cases are $1$, the empty product, and $-1$, a single reflection. Induction and reversal of the transformations prove the factorization.
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