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Householder Reflection
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
:
is obtained by
reflecting
about
This is called a
Householder reflection
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Download
Reverb.pdf
Download
Reverb_2up.pdf
Download
Reverb_4up.pdf
``
Artificial Reverberation and Spatialization
'', by Julius O. Smith III and Nelson Lee,
REALSIMPLE
Project — work supported by the
Wallenberg Global Learning Network
.
Released
2007-09-19
under the
Creative Commons License (Attribution 2.5)
, by Julius O. Smith III and Nelson Lee
Center for Computer Research in Music and Acoustics (CCRMA),
Stanford University