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