Gram–Schmidt process

related topics
{math, number, function}

In mathematics, particularly linear algebra and numerical analysis, the Gram–Schmidt process is a method for orthonormalising a set of vectors in an inner product space, most commonly the Euclidean space Rn. The Gram–Schmidt process takes a finite, linearly independent set S = {v1, …, vk} for kn and generates an orthogonal set S′ = {u1, …, uk} that spans the same k-dimensional subspace of Rn as S.

The method is named for Jørgen Pedersen Gram and Erhard Schmidt but it appeared earlier in the work of Laplace and Cauchy. 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 matrix yields the QR decomposition (it is decomposed into an orthogonal and a triangular matrix).


The Gram–Schmidt process

We define the projection operator by

where 〈u, v〉 denotes the inner product of the vectors u and v. This operator projects the vector v orthogonally onto the vector u.

The Gram–Schmidt process then works as follows:

The sequence u1, ..., uk is the required system of orthogonal vectors, and the normalized vectors e1, ..., ek form an orthonormal set. The calculation of the sequence u1, ..., uk is known as Gram–Schmidt orthogonalization, while the calculation of the sequence e1, ..., ek is known as Gram–Schmidt orthonormalization as the vectors are normalized.

Full article ▸

related documents
Open set
Depth-first search
Recursive descent parser
Compactification (mathematics)
Riesz representation theorem
Augmented Backus–Naur Form
Closure (topology)
Linear search
Procedural programming
Commutator subgroup
Fixed point combinator
Multiplicative function
Paracompact space
Prim's algorithm
Legendre polynomials
Hyperbolic function
Constructible number
Existential quantification
Jules Richard
Union (set theory)
Rank (linear algebra)
Riemann mapping theorem
Compactness theorem
Outer product
Integral domain
Bolzano–Weierstrass theorem
Poisson process