Householder factorization of an orthogonal matrix ($Q=H_1\cdots H_r,r\le n$)

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 $\mathrm{diag}(1,Q^{\prime })$, 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.