Editing Gram–Schmidt process (section)

{{Short description|Orthonormalization of a set of vectors}}
[[File:Gram–Schmidt process.svg|right|frame|The first two steps of the Gram–Schmidt process]]
In [[mathematics]], particularly [[linear algebra]] and [[numerical analysis]], the '''Gram–Schmidt process''' or Gram-Schmidt algorithm is a way of finding a set of two or more vectors that are perpendicular to each other.

By technical definition, it is a method of constructing an [[orthonormal basis]] from a set of [[vector (geometry)|vectors]] in an [[inner product space]], most commonly the [[Euclidean space]] <math>\mathbb{R}^n</math> equipped with the [[standard inner product]]. The Gram–Schmidt process takes a [[finite set|finite]], [[linearly independent]] set of vectors <math>S = \{ \mathbf{v}_1, \ldots , \mathbf{v}_k \}</math> for {{math|''k'' ≤ ''n''}} and generates an [[orthogonal set]]  <math>S' = \{ \mathbf{u}_1 , \ldots , \mathbf{u}_k \}</math> that spans the same <math>k</math>-dimensional subspace of <math>\mathbb{R}^n</math> as <math>S</math>.

The method is named after [[Jørgen Pedersen Gram]] and [[Erhard Schmidt]], but [[Pierre-Simon Laplace]] had been familiar with it before Gram and Schmidt.<ref>{{cite book |last1=Cheney |first1=Ward |last2=Kincaid |first2=David |title=Linear Algebra: Theory and Applications |location=Sudbury, Ma |publisher=Jones and Bartlett |year=2009 |url={{Google books |plainurl=yes |id=Gg3Uj1GkHK8C |page=544 }} |isbn=978-0-7637-5020-6 |pages=544, 558 }}</ref> In the theory of [[Lie group decompositions]], it is generalized by the [[Iwasawa decomposition]].

The application of the Gram–Schmidt process to the column vectors of a full column [[rank (linear algebra)|rank]] [[matrix (mathematics)|matrix]] yields the [[QR decomposition]] (it is decomposed into an [[orthogonal matrix|orthogonal]] and a [[triangular matrix]]).