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Householder transformation
This entry describes the Householder transformation $u=Hv$, the most frequently used algorithm for performing QR decomposition. The key object here is the Householder matrix $H$, a symmetric and orthogonal matrix of the form
$H=I2xx^{T},$ 
where $I$ is the identity matrix and we have used any normalized vector $x$ with $x_{2}^{2}=x^{T}x=1$.
The Householder transformation zeroes the last $m1$ elements of a column vector below the first element:
$\begin{bmatrix}v_{1}\\ v_{2}\\ \vdots\\ v_{m}\end{bmatrix}\rightarrow\begin{bmatrix}c\\ 0\\ \vdots\\ 0\end{bmatrix}\text{with}\;c=\pmv_{2}=\pm\left(\sum_{{i=1}}^{m}v_{i}^{2}% \right)^{{1/2}}$ 
One can verify that
$x=f\begin{bmatrix}v_{1}c\\ v_{2}\\ \vdots\\ v_{m}\end{bmatrix}\text{with}\;f=\frac{1}{\sqrt{2c(cv_{1})}}$ 
fulfils $x^{T}x=1$ and that with $H=I2xx^{T}$ one obtains the vector $\begin{bmatrix}c&0&\cdots&0\end{bmatrix}^{T}$.
To perform the decomposition of the $m\times n$ matrix $A=QR$ (with $m\geq n$) we construct an $m\times m$ matrix $H^{{(1)}}$ to change the $m1$ elements of the first column to zero. Similarly, an $m1\times m1$ matrix $G^{{(2)}}$ will change the $m2$ elements of the second column to zero. With $G^{{(2)}}$ we produce the $m\times m$ matrix
$H^{{(2)}}=\begin{bmatrix}1&0&\cdots&0\\ 0&&&\\ \vdots&&G^{{(2)}}&\\ 0&&&\end{bmatrix}.$ 
After $n$ such orthogonal transformations ($n1$ times in the case that $m=n$), we let
$R=H^{{(n)}}\cdots H^{{(2)}}H^{{(1)}}A.$ 
$R$ is upper triangular and the orthogonal matrix $Q$ becomes
$Q=H^{{(1)}}H^{{(2)}}\cdots H^{{(n)}}.$ 
In practice the $H^{{(i)}}$ are never explicitly computed.
References

Originally from The Data Analysis Briefbook (http://rkb.home.cern.ch/rkb/titleA.html)
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