GrammSchmidtprocessen

The Gram-Schmidt process is a method for orthonormalizing a set of vectors in an inner product space, most commonly the Euclidean space. It was developed by Erhard Schmidt in 1907, based on earlier work by Joseph-Louis Lagrange and later by Paul du Bois-Reymond. The process is widely used in linear algebra, numerical analysis, and various applications in mathematics and physics.

The Gram-Schmidt process takes a set of linearly independent vectors and produces an orthonormal set of vectors. An orthonormal set is a set of vectors that are mutually orthogonal (perpendicular) and have unit length. This process is particularly useful in constructing orthogonal bases for vector spaces.

The algorithm works as follows: given a set of linearly independent vectors {v1, v2, ..., vn}, the process constructs an orthonormal set {u1, u2, ..., un} such that the span of {u1, u2, ..., uk} is the same as the span of {v1, v2, ..., vk} for each k.

The first vector u1 is simply the normalization of v1. For subsequent vectors, each vector uk is obtained by subtracting the projections of uk onto all previously obtained vectors {u1, u2, ..., uk-1} from vk, and then normalizing the result.

The Gram-Schmidt process is not numerically stable for large sets of vectors due to the accumulation of rounding errors. More stable algorithms, such as the modified Gram-Schmidt process and the Householder transformations, are often used in practice. Despite its numerical instability, the Gram-Schmidt process remains an important theoretical tool in linear algebra.