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For completeness, this section derives the Householder reflection
matrix from geometric considerations [454]. 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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