Orthogonalisointi

Orthogonalisointi, meaning orthogonalization in English, is a process used in linear algebra and related fields to transform a set of vectors into a set of orthogonal vectors. Orthogonal vectors are vectors that are perpendicular to each other, meaning their dot product is zero. The most well-known method for performing orthogonalization is the Gram-Schmidt process.

The Gram-Schmidt process takes a set of linearly independent vectors and produces an orthogonal set of vectors that span the same subspace. The process is iterative. It starts with the first vector and then, for each subsequent vector, it subtracts its projection onto the subspace spanned by the previously orthogonalized vectors. This ensures that the resulting vector is orthogonal to all previous vectors.

Orthogonalization has numerous applications. It is fundamental in numerical analysis for solving systems of linear equations and for performing least-squares approximations. In signal processing, orthogonal bases are used to decompose signals into components that are independent of each other, which simplifies analysis and processing. In quantum mechanics, orthogonal states represent distinct physical configurations. The concept is also crucial in areas like machine learning, particularly in algorithms involving dimensionality reduction and feature extraction, such as Principal Component Analysis (PCA).