This entry describes the Householder transformation , the most frequently used algorithm for performing QR decomposition. The key object here is the Householder matrix , a symmetric and orthogonal matrix of the form
The Householder transformation zeroes the last elements of a column vector below the first element:
One can verify that
fulfils and that with one obtains the vector .
To perform the decomposition of the matrix (with ) we construct an matrix to change the elements of the first column to zero. Similarly, an matrix will change the elements of the second column to zero. With we produce the matrix
After such orthogonal transformations ( times in the case that ), we let
is upper triangular and the orthogonal matrix becomes
In practice the are never explicitly computed.
Originally from The Data Analysis Briefbook (http://rkb.home.cern.ch/rkb/titleA.htmlhttp://rkb.home.cern.ch/rkb/titleA.html)
|Date of creation||2013-03-22 12:06:07|
|Last modified on||2013-03-22 12:06:07|
|Last modified by||mathcam (2727)|