Gram-Schmidt orthogonalization
Any set of linearly independent vectors can be converted into a set of orthogonal vectors by the Gram-Schmidt process. In three dimensions, determines a line; the vectors and determine a plane. The vector is the unit vector in the direction . The (unit) vector lies in the plane of , and is normal to (on the same side as . The (unit) vector is normal to the plane of , on the same side as , etc.
In general, first set , and then each is made orthogonal to the preceding by subtraction of the projections of in the directions of :
The vectors span the same subspace as the . The vectors are orthonormal. This leads to the following theorem:
Theorem.
Any matrix with linearly independent columns can be factorized into a product, . The columns of are orthonormal and is upper triangular and invertible.
This “classical” Gram-Schmidt method is often numerically unstable, see [Golub89] for a “modified” Gram-Schmidt method.
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Originally from The Data Analysis Briefbook (http://rkb.home.cern.ch/rkb/titleA.htmlhttp://rkb.home.cern.ch/rkb/titleA.html)
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Golub89
Gene H. Golub and Charles F. van Loan: Matrix Computations, 2nd edn., The John Hopkins University Press, 1989.
Title | Gram-Schmidt orthogonalization |
Canonical name | GramSchmidtOrthogonalization |
Date of creation | 2013-03-22 12:06:14 |
Last modified on | 2013-03-22 12:06:14 |
Owner | akrowne (2) |
Last modified by | akrowne (2) |
Numerical id | 9 |
Author | akrowne (2) |
Entry type | Algorithm |
Classification | msc 65F25 |
Synonym | Gram-Schmidt decomposition |
Synonym | Gram-Schmidt orthonormalization |
Synonym | Gram-Schmidt process |
Related topic | HouseholderTransformation |
Related topic | GivensRotation |
Related topic | QRDecomposition |
Related topic | AnExampleForSchurDecomposition |