Gram–Schmidt process

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 equipped with the standard inner product. The Gram–Schmidt process takes a finite, linearly independent set S = {v₁, ..., v_k} for k ≤ n and generates an orthogonal set S′ = {u₁, ..., u_k} that spans the same k-dimensional subspace of Rⁿ as S.

The method is named after 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).

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