orthogonisering

Orthogonalisering, or orthogonalization, is the process of transforming a set of vectors into an orthogonal (or orthonormal) set with respect to a given inner product. In linear algebra, an orthogonal collection consists of vectors whose pairwise inner products are zero; an orthonormal set consists of vectors that are both orthogonal and of unit length.

The most common method is the Gram–Schmidt process. Given a sequence of linearly independent vectors, Gram–Schmidt constructs an orthogonal (or orthonormal) sequence by iteratively subtracting the projections of each new vector onto the previously obtained vectors, and then normalizing if an orthonormal set is required. In complex spaces, inner products involve complex conjugation, which is accounted for in the projection step.

Alternative, more numerically stable methods include Householder reflections and Givens rotations. These techniques underpin robust QR factorizations, where a matrix is decomposed into an orthogonal (or unitary) matrix and an upper triangular matrix. Such factorizations are widely used in solving linear systems, least squares problems, and eigenvalue computations.

Key properties include that orthogonalization of a linearly independent set yields an orthonormal basis for the same span, though the resulting vectors depend on the order in which the original vectors are processed. In infinite-dimensional Hilbert spaces, Gram–Schmidt produces an orthonormal system that spans the closure of the given subspace.

Applications of orthogonalisering span numerical linear algebra, signal processing, statistics (notably principal component analysis), and data compression. In practice, numerical stability and re-orthogonalization considerations often guide the choice of algorithm.