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2.2.2 Householder Transformation
Task: Strip column and/or row vectors of some of their rear elements.
Given
triangular, tridiagonal, or otherwise simple.
Let be a vector (such as a matrix column vector),
and a unit vector. Define an auxiliary vector
and normalize it.
Householder matrix:
See for yourself that in the transformed vector
all elements but the first are equal to zero.
Apply this transformation successively to simplify a matrix.
EXAMPLE:
Let
From
construct the auxiliary vector
, normalize to
.
The Householder matrix is
Check: Multiplying
by
we have
Next step: treat the lower right submatrix of
. From
construct a
Householder matrix which is promoted to a matrix by
adding a trivial first line and column:
Result:
Next: 2.2.3 LU Decomposition
Up: 2.2 Exact Methods
Previous: 2.2.1 Gauss Elimination and
Franz J. Vesely Oct 2005
See also: "Computational Physics - An Introduction," Kluwer-Plenum 2001