After a permutation of columns (if necessary) any
matrix
can be factored into a product of a unitary and a generalized
upper triangular matrix
This can be accomplished by
Householder transformations at most:
Before the -th transformation the modified matrix
has the form
If is zero, the upper triangular form is reached. Otherwise, two
columns are interchanged in such a way that the first column
of has the largest 2-norm.
Then, a Householder transformation based on is applied to ,
advancing the triangular structure one step further.
The permutations are recorded in the index vector , which initially
is set to
. For any column interchange
the corresponding indices in
are permuted.
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7/ 2/2007 |