In linear algebra, orthogonalization is the process of finding a set of orthogonal vectors that span a particular subspace. Formally, starting with a linearly independent set of vectors {v1, ... , vk} in an inner product space (most commonly the Euclidean spaceRn), orthogonalization results in a set of orthogonal vectors {u1, ... , uk} that generate the same subspace as the vectors v1, ... , vk. Every vector in the new set is orthogonal to every other vector in the new set; and the new set and the old set have the same linear span.
In addition, if we want the resulting vectors to all be unit vectors, then we normalize each vector and the procedure is called orthonormalization.
Orthogonalization is also possible with respect to any symmetric bilinear form (not necessarily an inner product, not necessarily over real numbers), but standard algorithms may encounter division by zero in this more general setting.
When performing orthogonalization on a computer, the Householder transformation is usually preferred over the Gram–Schmidt process since it is more numerically stable, i.e. rounding errors tend to have less serious effects.
On the other hand, the Gram–Schmidt process produces the jth orthogonalized vector after the jth iteration, while orthogonalization using Householder reflections produces all the vectors only at the end. This makes only the Gram–Schmidt process applicable for iterative methods like the Arnoldi iteration.
The Givens rotation is more easily parallelized than Householder transformations.
To compensate for the loss of useful signal in traditional noise attenuation approaches because of incorrect parameter selection or inadequacy of denoising assumptions, a weighting operator can be applied on the initially denoised section for the retrieval of useful signal from the initial noise section. The new denoising process is referred to as the local orthogonalization of signal and noise.[2] It has a wide range of applications in many signals processing and seismic exploration fields.
See alsoedit
Look up orthogonalization in Wiktionary, the free dictionary.
^Löwdin, Per-Olov (1970). "On the nonorthogonality problem". Advances in quantum chemistry. Vol. 5. Elsevier. pp. 185–199. doi:10.1016/S0065-3276(08)60339-1. ISBN9780120348053.
^Chen, Yangkang; Fomel, Sergey (2015). "Random noise attenuation using local signal-and-noise orthogonalization". Geophysics. 80 (6): WD1–WD9. Bibcode:2015Geop...80D...1C. doi:10.1190/GEO2014-0227.1.
January 01, 1970
orthogonalization, this, article, needs, additional, citations, verification, please, help, improve, this, article, adding, citations, reliable, sources, unsourced, material, challenged, removed, find, sources, news, newspapers, books, scholar, jstor, january,. This article needs additional citations for verification Please help improve this article by adding citations to reliable sources Unsourced material may be challenged and removed Find sources Orthogonalization news newspapers books scholar JSTOR January 2021 Learn how and when to remove this message In linear algebra orthogonalization is the process of finding a set of orthogonal vectors that span a particular subspace Formally starting with a linearly independent set of vectors v1 vk in an inner product space most commonly the Euclidean space Rn orthogonalization results in a set of orthogonal vectors u1 uk that generate the same subspace as the vectors v1 vk Every vector in the new set is orthogonal to every other vector in the new set and the new set and the old set have the same linear span In addition if we want the resulting vectors to all be unit vectors then we normalize each vector and the procedure is called orthonormalization Orthogonalization is also possible with respect to any symmetric bilinear form not necessarily an inner product not necessarily over real numbers but standard algorithms may encounter division by zero in this more general setting Contents 1 Orthogonalization algorithms 2 Local orthogonalization 3 See also 4 ReferencesOrthogonalization algorithms editMethods for performing orthogonalization include Gram Schmidt process which uses projection Householder transformation which uses reflection Givens rotation Symmetric orthogonalization which uses the Singular value decomposition When performing orthogonalization on a computer the Householder transformation is usually preferred over the Gram Schmidt process since it is more numerically stable i e rounding errors tend to have less serious effects On the other hand the Gram Schmidt process produces the jth orthogonalized vector after the jth iteration while orthogonalization using Householder reflections produces all the vectors only at the end This makes only the Gram Schmidt process applicable for iterative methods like the Arnoldi iteration The Givens rotation is more easily parallelized than Householder transformations Symmetric orthogonalization was formulated by Per Olov Lowdin 1 Local orthogonalization editTo compensate for the loss of useful signal in traditional noise attenuation approaches because of incorrect parameter selection or inadequacy of denoising assumptions a weighting operator can be applied on the initially denoised section for the retrieval of useful signal from the initial noise section The new denoising process is referred to as the local orthogonalization of signal and noise 2 It has a wide range of applications in many signals processing and seismic exploration fields See also edit nbsp Look up orthogonalization in Wiktionary the free dictionary Orthogonality Biorthogonal system Orthogonal basisReferences edit Lowdin Per Olov 1970 On the nonorthogonality problem Advances in quantum chemistry Vol 5 Elsevier pp 185 199 doi 10 1016 S0065 3276 08 60339 1 ISBN 9780120348053 Chen Yangkang Fomel Sergey 2015 Random noise attenuation using local signal and noise orthogonalization Geophysics 80 6 WD1 WD9 Bibcode 2015Geop 80D 1C doi 10 1190 GEO2014 0227 1 Retrieved from https en wikipedia org w index php title Orthogonalization amp oldid 1196545301, wikipedia, wiki, book, books, library,
