Ortonormalisation

Ortonormalisation is a process in linear algebra used to transform a set of linearly independent vectors into an orthonormal set. An orthonormal set of vectors is one where each vector has a magnitude (or norm) of one, and every distinct pair of vectors in the set is orthogonal, meaning their dot product is zero.

The most common method for achieving ortonormalisation is the Gram-Schmidt process. This iterative algorithm takes a set of linearly independent vectors as input and produces an orthonormal set spanning the same subspace. The process involves taking each vector in the original set, projecting it onto the subspace spanned by the previously orthonormalized vectors, and then subtracting this projection from the original vector. This resulting vector is then orthogonal to all previously generated orthonormal vectors. Finally, this orthogonal vector is normalized by dividing it by its magnitude to make it a unit vector.

Orthonormal sets have many important applications in mathematics, physics, and engineering. They simplify many calculations and are fundamental to concepts like Fourier series, QR decomposition, and principal component analysis. The ability to represent vectors in terms of an orthonormal basis is particularly useful for decomposing complex problems into simpler, independent components. The process ensures that each component contributes independently to the overall solution.