Next  |  Prev  |  Up  |  Top  |  Index  |  JOS Index  |  JOS Pubs  |  JOS Home  |  Search

Householder Reflections

For completeness, this section derives the Householder reflection matrix from geometric considerations [454]. Let $ \mathbf{P}_{\underline{u}}$ denote the projection matrix which orthogonally projects vectors onto $ {\underline{u}}$ , i.e.,

$\displaystyle \mathbf{P}_{\underline{u}}= \frac{\underline{u}\,\underline{u}^T}{\underline{u}^T\underline{u}} = \frac{\underline{u}\,\underline{u}^T}{\left\Vert\,\underline{u}\,\right\Vert^2}
$

and

$\displaystyle \mathbf{P}_{\underline{u}}\, \underline{x}= \underline{u}\,\frac{\left<\underline{u},\underline{x}\right>}{\left\Vert\,\underline{u}\,\right\Vert^2}
$

specifically projects $ \underline {x}$ onto $ \underline{u}$ . Since the projection is orthogonal, we have

$\displaystyle \left<\underline{x}-\mathbf{P}_{\underline{u}}\underline{x},\underline{u}\right>=\left<(\mathbf{I}-\mathbf{P}_{\underline{u}})\underline{x},\underline{u}\right>=\underline{0}.
$

We may interpret $ (\mathbf{I}-\mathbf{P}_{\underline{u}})\underline{x}$ as the difference vector between $ \underline {x}$ and $ \mathbf{P}_{\underline{u}}\underline{x}$ , its orthogonal projection onto $ \underline{u}$ , since

$\displaystyle (\mathbf{I}-\mathbf{P}_{\underline{u}})\underline{x}+ \mathbf{P}_{\underline{u}}\underline{x}= \underline{x}
$

and we have $ (\mathbf{I}-\mathbf{P}_{\underline{u}})\underline{x}\perp \mathbf{P}_{\underline{u}}\underline{x}$ by definition of the orthogonal projection. Consequently, the projection onto $ \underline{u}$ minus this difference vector gives a reflection of the vector $ \underline {x}$ about $ \underline{u}$ :

$\displaystyle \underline{y}= \mathbf{P}_{\underline{u}}\underline{x}- (\mathbf{I}-\mathbf{P}_{\underline{u}})\underline{x}= (2\mathbf{P}_{\underline{u}}- \mathbf{I})\underline{x}
$

Thus, $ \underline{y}$ is obtained by reflecting $ \underline {x}$ about $ \underline{u}$ --a so-called Householder reflection.


Next  |  Prev  |  Up  |  Top  |  Index  |  JOS Index  |  JOS Pubs  |  JOS Home  |  Search

[How to cite this work]  [Order a printed hardcopy]  [Comment on this page via email]

``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4
Copyright © 2024-06-28 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
CCRMA