#### Householder Reflections

For completeness, this section derives the Householder reflection matrix from geometric considerations [451]. Let denote the projection matrix which orthogonally projects vectors onto , i.e.,

and

specifically projects onto . Since the projection is orthogonal, we have

We may interpret as the difference vector between and , its orthogonal projection onto , since

and we have by definition of the orthogonal projection. Consequently, the projection onto minus this difference vector gives a reflection of the vector about :

Thus, is obtained by reflecting about --a so-called Householder reflection.
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