Editing Singular value decomposition (section)

==History==
The singular value decomposition was originally developed by [[differential geometry|differential geometers]], who wished to determine whether a real [[bilinear form]] could be made equal to another by independent orthogonal transformations of the two spaces it acts on. [[Eugenio Beltrami]] and [[Camille Jordan]] discovered independently, in 1873 and 1874 respectively, that the singular values of the bilinear forms, represented as a matrix, form a [[Complete set of invariants|complete set]] of [[invariant (mathematics)|invariant]]s for bilinear forms under orthogonal substitutions. [[James Joseph Sylvester]] also arrived at the singular value decomposition for real square matrices in 1889, apparently independently of both Beltrami and Jordan. Sylvester called the singular values the ''canonical multipliers'' of the matrix {{tmath|\mathbf A.}} The fourth mathematician to discover the singular value decomposition independently is [[Léon Autonne|Autonne]] in 1915, who arrived at it via the [[polar decomposition]]. The first proof of the singular value decomposition for rectangular and complex matrices seems to be by [[Carl Eckart]] and [[Gale J. Young]] in 1936;<ref>{{Cite journal
 |last1=Eckart |first1=C.|author-link1=Carl Eckart |last2=Young |first2=G. |year=1936  |title=The approximation of one matrix by another of lower rank |journal=[[Psychometrika]] |volume=1 |issue=3 |pages=211–8  |doi=10.1007/BF02288367  |s2cid=10163399}}</ref> they saw it as a generalization of the [[Principal axis theorem|principal axis]] transformation for [[Hermitian matrix|Hermitian matrices]].

In 1907, [[Erhard Schmidt]] defined an analog of singular values for [[integral operator]]s (which are compact, under some weak technical assumptions); it seems he was unaware of the parallel work on singular values of finite matrices. This theory was further developed by [[Émile Picard]] in 1910, who is the first to call the numbers <math>\sigma_k</math> ''singular values'' (or in French, ''valeurs singulières'').

Practical methods for computing the SVD date back to [[Ervand Kogbetliantz|Kogbetliantz]] in 1954–1955 and [[Magnus Hestenes|Hestenes]] in 1958,<ref>{{Cite journal |first=M. R. |last=Hestenes |author-link=Magnus Hestenes
 |title=Inversion of Matrices by Biorthogonalization and Related Results
 |journal=Journal of the Society for Industrial and Applied Mathematics
 |year=1958 |volume=6 |issue=1 |pages=51–90
 |doi=10.1137/0106005 |mr=0092215 | jstor = 2098862
 }}</ref> resembling closely the [[Jacobi eigenvalue algorithm]], which uses plane rotations or [[Givens rotation]]s. However, these were replaced by the method of [[Gene H. Golub|Gene Golub]] and [[William Kahan]] published in 1965,<ref>{{harv|Golub|Kahan|1965}}</ref> which uses [[Householder transformation]]s or reflections. In 1970, Golub and [[Christian Reinsch]]<ref>{{Cite journal
 |title=Singular value decomposition and least squares solutions
 |first1=G. H. |last1=Golub |author-link1=Gene H. Golub
 |first2=C. |last2=Reinsch|author2-link=Christian Reinsch 
 |year=1970
 |journal=Numerische Mathematik
 |volume=14 |issue=5 |pages=403–420
 |doi=10.1007/BF02163027 |mr=1553974
 |s2cid=123532178 
 }}</ref> published a variant of the Golub/Kahan algorithm that is still the one most-used today.